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Orthogonal matrix
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For matrices with orthogonality over the complex number field, see unitary matrix.
In linear algebra, an orthogonal matrix or real orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.
Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}
where I {\displaystyle I} is the identity matrix.
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
Q T = Q − 1 . {\displaystyle Q^{\mathrm {T} }=Q^{-1}.}
An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗) and therefore normal (Q∗Q = QQ∗) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.
The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.
Contents
[hide]
1 Overview
2 Examples
3 Elementary constructions
3.1 Lower dimensions
3.2 Higher dimensions
3.3 Primitives
4 Properties
4.1 Matrix properties
4.2 Group properties
4.3 Canonical form
4.4 Lie algebra
5 Numerical linear algebra
5.1 Benefits
5.2 Decompositions
5.2.1 Examples
5.3 Randomization
5.4 Nearest orthogonal matrix
6 Spin and pin
7 Rectangular matrices
8 See also
9 Notes
10 References
11 External links
Overview[edit]
An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,[1] so, for vectors u and v in an n-dimensional real Euclidean space
u ⋅ v = ( Q u ) ⋅ ( Q v ) {\displaystyle {\mathbf {u} }\cdot {\mathbf {v} }=\left(Q{\mathbf {u} }\right)\cdot \left(Q{\mathbf {v} }\right)\,}
where Q is an orthogonal matrix. To see the inner product connection, consider a vector v in an n-dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of v is vTv. If a linear transformation, in matrix form Qv, preserves vector lengths, then
v T v = ( Q v ) T ( Q v ) = v T Q T Q v . {\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }.}
Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as QR decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.
Examples[edit]
Below are a few examples of small orthogonal matrices and possible interpretations.
[ 1 0 0 1 ] ( identity transformation ) {\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad ({\text{identity transformation}})}
An instance of a 2 × 2 rotation matrix:
R ( 16.26 ∘ ) = [ cos  θ − sin  θ sin  θ cos  θ ] = [ 0.96 − 0.28 0.28 0.96 ] ( rotation by 16.26 ∘ ) {\displaystyle R(16.26^{\circ })={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}={\begin{bmatrix}0.96&-0.28\\0.28&\;\;\,0.96\\\end{bmatrix}}\qquad ({\text{rotation by }}16.26^{\circ })}
[ 1 0 0 − 1 ] ( reflection across x -axis ) {\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}\qquad ({\text{reflection across }}x{\text{-axis}})}
[ 0 − 0.80 − 0.60 0.80 − 0.36 0.48 0.60 0.48 − 0.64 ] ( rotoinversion: axis ( 0 , − 3 5 , 4 5 ) , angle 90 ∘ ) {\displaystyle {\begin{bmatrix}0&-0.80&-0.60\\0.80&-0.36&\;\;\,0.48\\0.60&\;\;\,0.48&-0.64\end{bmatrix}}\qquad \left({\begin{aligned}&{\text{rotoinversion:}}\\&{\text{axis }}\left(0,-{\tfrac {3}{5}},{\tfrac {4}{5}}\right),{\text{ angle }}90^{\circ }\end{aligned}}\right)}
[ 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 ] ( permutation of coordinate axes ) {\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}\qquad ({\text{permutation of coordinate axes}})}
Elementary constructions[edit]
Lower dimensions[edit]
The simplest orthogonal matrices are the 1 × 1 matrices [1] and [−1] which we can interpret as the identity and a reflection of the real line across the origin.
The 2 × 2 matrices have the form
[ p t q u ] , {\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}
which orthogonality demands satisfy the three equations
1 = p 2 + t 2 , 1 = q 2 + u 2 , 0 = p q + t u . {\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\end{aligned}}}
In consideration of the first equation, without loss of generality let p = cos θ, q = sin θ; then either t = −q, u = p or t = q, u = −p. We can interpret the first case as a rotation by θ (where θ = 0 is the identity), and the second as a reflection across a line at an angle of θ/2.
[ cos  θ − sin  θ sin  θ cos  θ ] (rotation), [ cos  θ sin  θ sin  θ − cos  θ ] (reflection) {\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}
The special case of the reflection matrix with θ = 90° generates a reflection about the line at 45° given by y = x and therefore exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):
[ 0 1 1 0 ] . {\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}
The identity is also a permutation matrix.
A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.
Higher dimensions[edit]
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3 × 3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
[ − 1 0 0 0 − 1 0 0 0 − 1 ] and [ 0 − 1 0 1 0 0 0 0 − 1 ] {\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}
represent an inversion through the origin and a rotoinversion, respectively, about the z-axis.
[ cos  α cos  γ − sin  α sin  β sin  γ − sin  α cos  β − cos  α sin  γ − sin  α sin  β cos  γ cos  α sin  β sin  γ + sin  α cos  γ cos  α cos  β cos  α sin  β cos  γ − sin  α sin  γ cos  β sin  γ − sin  β cos  β cos  γ ] {\displaystyle {\begin{bmatrix}\cos \alpha \cos \gamma -\sin \alpha \sin \beta \sin \gamma &-\sin \alpha \cos \beta &-\cos \alpha \sin \gamma -\sin \alpha \sin \beta \cos \gamma \\\cos \alpha \sin \beta \sin \gamma +\sin \alpha \cos \gamma &\cos \alpha \cos \beta &\cos \alpha \sin \beta \cos \gamma -\sin \alpha \sin \gamma \\\cos \beta \sin \gamma &-\sin \beta &\cos \beta \cos \gamma \end{bmatrix}}}
Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a 3 × 3 rotation matrix in terms of an axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a plane of rotation.
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
Primitives[edit]
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any n × n permutation matrix can be constructed as a product of no more than n − 1 transpositions.
A Householder reflection is constructed from a non-null vector v as
Q = I − 2 v v T v T v . {\displaystyle Q=I-2{\frac {{\mathbf {v} }{\mathbf {v} }^{\mathrm {T} }}{{\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }}}.}
Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vvT suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.
A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size n × n can be constructed as a product of at most n(n − 1)/2 such rotations. In the case of 3 × 3 matrices, three such rotations suffice; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles.
A Jacobi rotation has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a 2 × 2 symmetric submatrix.
Properties[edit]
Matrix properties[edit]
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space ℝn with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of ℝn. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy MTM = D, with D a diagonal matrix.
The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:
1 = det ( I ) = det ( Q T Q ) = det ( Q T ) det ( Q ) = ( det ( Q ) ) 2 . {\displaystyle 1=\det(I)=\det \left(Q^{\mathrm {T} }Q\right)=\det \left(Q^{\mathrm {T} }\right)\det(Q)={\bigl (}\det(Q){\bigr )}^{2}.}
The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.
[ 2 0 0 1 2 ] {\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}
With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.
Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.
Group properties[edit]
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all n × n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension n(n − 1)/2, called the orthogonal group and denoted by O(n).
The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations. The quotient group O(n)/SO(n) is isomorphic to O(1), with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2 × 2 matrices. If n is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.
Now consider (n + 1) × (n + 1) orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an n × n orthogonal matrix; thus O(n) is a subgroup of O(n + 1) (and of all higher groups).
[ 0 O ( n ) ⋮ 0 0 ⋯ 0 1 ] {\displaystyle {\begin{bmatrix}&&&0\\&\mathrm {O} (n)&&\vdots \\&&&0\\0&\cdots &0&1\end{bmatrix}}}
Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of O(n) in O(n + 1); in this way O(n + 1) is a bundle over the unit sphere Sn with fiber O(n).
Similarly, SO(n) is a subgroup of SO(n + 1); and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: SO(n) ↪ SO(n + 1) → Sn. A single rotation can produce a zero in the first row of the last column, and series of n − 1 rotations will zero all but the last row of the last column of an n × n rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, SO(n) therefore has
( n − 1 ) + ( n − 2 ) + ⋯ + 1 = n ( n − 1 ) 2 {\displaystyle (n-1)+(n-2)+\cdots +1={\frac {n(n-1)}{2}}}
degrees of freedom, and so does O(n).
Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order n! symmetric group Sn. By the same kind of argument, Sn is a subgroup of Sn + 1. The even permutations produce the subgroup of permutation matrices of determinant +1, the order n!/2 alternating group.
Canonical form[edit]
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if Q is special orthogonal then one can always find an orthogonal matrix P, a (rotational) change of basis, that brings Q into block diagonal form:
P T Q P = [ R 1 ⋱ R k ] ( n even ) , P T Q P = [ R 1 ⋱ R k 1 ] ( n odd ) . {\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}
where the matrices R1, ..., Rk are 2 × 2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, ±I. Thus, negating one column if necessary, and noting that a 2 × 2 reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form
P T Q P = [ R 1 ⋱ R k 0 0 ± 1 ⋱ ± 1 ] , {\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}
The matrices R1, ..., Rk give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If n is odd, there is at least one real eigenvalue, +1 or −1; for a 3 × 3 rotation, the eigenvector associated with +1 is the rotation axis.
Lie algebra[edit]
Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition
Q T Q = I {\displaystyle Q^{\mathrm {T} }Q=I}
yields
Q ˙ T Q + Q T Q ˙ = 0 {\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}
Evaluation at t = 0 (Q = I) then implies
Q ˙ T = − Q ˙ . {\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}
In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).
For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra s o {\displaystyle {\mathfrak {so}}} (3) tangent to SO(3). Given ω = (xθ, yθ, zθ), with v = (x, y, z) being a unit vector, the correct skew-symmetric matrix form of ω is
Ω = [ 0 − z θ y θ z θ 0 − x θ − y θ x θ 0 ] . {\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}
The exponential of this is the orthogonal matrix for rotation around axis v by angle θ; setting c = cos θ/2, s = sin θ/2,
exp  ( Ω ) = [ 1 − 2 s 2 + 2 x 2 s 2 2 x y s 2 − 2 z s c 2 x z s 2 + 2 y s c 2 x y s 2 + 2 z s c 1 − 2 s 2 + 2 y 2 s 2 2 y z s 2 − 2 x s c 2 x z s 2 − 2 y s c 2 y z s 2 + 2 x s c 1 − 2 s 2 + 2 z 2 s 2 ] . {\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}
Numerical linear algebra[edit]
Benefits[edit]
Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.
Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of n indices.
Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order n3 to a much more efficient order n. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following Stewart (1976), we do not store a rotation angle, which is both expensive and badly behaved.)
Decompositions[edit]
A number of important matrix decompositions (Golub & Van Loan 1996) involve orthogonal matrices, including especially:
QR decomposition
M = QR, Q orthogonal, R upper triangular
Singular value decomposition
M = UΣVT, U and V orthogonal, Σ diagonal matrix
Eigendecomposition of a symmetric matrix (decomposition according to the spectral theorem)
S = QΛQT, S symmetric, Q orthogonal, Λ diagonal
Polar decomposition
M = QS, Q orthogonal, S symmetric non-negative definite
Examples[edit]
Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write Ax = b, where A is m × n, m > n. A QR decomposition reduces A to upper triangular R. For example, if A is 5 × 3 then R has the form
R = [ ⋅ ⋅ ⋅ 0 ⋅ ⋅ 0 0 ⋅ 0 0 0 0 0 0 ] . {\displaystyle R={\begin{bmatrix}\cdot &\cdot &\cdot \\0&\cdot &\cdot \\0&0&\cdot \\0&0&0\\0&0&0\end{bmatrix}}.}
The linear least squares problem is to find the x that minimizes ||Ax − b||, which is equivalent to projecting b to the subspace spanned by the columns of A. Assuming the columns of A (and hence R) are independent, the projection solution is found from ATAx = ATb. Now ATA is square (n × n) and invertible, and also equal to RTR. But the lower rows of zeros in R are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing ATA = (RTQT)QR to RTR, but also for allowing solution without magnifying numerical problems.
In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful. With A factored as UΣVT, a satisfactory solution uses the Moore-Penrose pseudoinverse, VΣ+UT, where Σ+ merely replaces each non-zero diagonal entry with its reciprocal. Set x to VΣ+UTb.
The case of a square invertible matrix also holds interest. Suppose, for example, that A is a 3 × 3 rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so A has gradually lost its true orthogonality. A Gram-Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton's method" approach due to Higham (1986) (1990), repeatedly averaging the matrix with its inverse transpose. Dubrulle (1994) has published an accelerated method with a convenient convergence test.
For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
[ 3 1 7 5 ] → [ 1.8125 0.0625 3.4375 2.6875 ] → ⋯ → [ 0.8 − 0.6 0.6 0.8 ] {\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
and which acceleration trims to two steps (with γ = 0.353553, 0.565685).
[ 3 1 7 5 ] → [ 1.41421 − 1.06066 1.06066 1.41421 ] → [ 0.8 − 0.6 0.6 0.8 ] {\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.
[ 3 1 7 5 ] → [ 0.393919 − 0.919145 0.919145 0.393919 ] {\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}
Randomization[edit]
Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformly distributed orthogonal matrices[citation needed], but the QR decomposition of independent normally distributed random entries does, as long as the diagonal of R contains only positive entries. Stewart (1980) replaced this with a more efficient idea that Diaconis & Shahshahani (1987) later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an (n + 1) × (n + 1) orthogonal matrix, take an n × n one and a uniformly distributed unit vector of dimension n + 1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).
Nearest orthogonal matrix[edit]
The problem of finding the orthogonal matrix Q nearest a given matrix M is related to the Orthogonal Procrustes problem. There are several different ways to get the unique solution, the simplest of which is taking the singular value decomposition of M and replacing the singular values with ones. Another method expresses the R explicitly but requires the use of a matrix square root:[2]
Q = M ( M T M ) − 1 2 {\displaystyle Q=M\left(M^{\mathrm {T} }M\right)^{-{\frac {1}{2}}}}
This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
Q n + 1 = 2 M ( Q n − 1 M + M T Q n ) − 1 {\displaystyle Q_{n+1}=2M\left(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n}\right)^{-1}}
where Q0 = M. These iterations are stable provided the condition number of M is less than three.[3]
Spin and pin[edit]
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected and thus the universal covering group for SO(n). By far the most famous example of a spin group is Spin(3), which is nothing but SU(2), or the group of unit quaternions.
The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.
Rectangular matrices[edit]
If Q is not a square matrix, then the conditions QTQ = I and QQT = I are not equivalent. The condition QTQ = I says that the columns of Q are orthonormal. This can only happen if Q is an m × n matrix with n ≤ m. Similarly, QQT = I says that the rows of Q are orthonormal, which requires n ≥ m.
There is no standard terminology for these matrices. They are sometimes called "orthonormal matrices", sometimes "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".
See also[edit]
Orthogonal group
Rotation (mathematics)
Skew-symmetric matrix, a matrix whose transpose is its negative
Symplectic matrix
Notes[edit]
Jump up ^ "Paul's online math notes"[full citation needed], Paul Dawkins, Lamar University, 2008. Theorem 3(c)
Jump up ^ "Finding the Nearest Orthonormal Matrix", Berthold K. P. Horn, MIT.
Jump up ^ "Newton's Method for the Matrix Square Root", Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.
References[edit]
Diaconis, Persi; Shahshahani, Mehrdad (1987), "The subgroup algorithm for generating uniform random variables", Prob. In Eng. And Info. Sci., 1: 15–32, doi:10.1017/S0269964800000255, ISSN 0269-9648
Dubrulle, Augustin A. (1999), "An Optimum Iteration for the Matrix Polar Decomposition", Elect. Trans. Num. Anal., 8: 21–25
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3/e ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9
Higham, Nicholas (1986), "Computing the Polar Decomposition—with Applications", SIAM Journal on Scientific and Statistical Computing, 7 (4): 1160–1174, doi:10.1137/0907079, ISSN 0196-5204
Higham, Nicholas; Schreiber, Robert (July 1990), "Fast polar decomposition of an arbitrary matrix", SIAM Journal on Scientific and Statistical Computing, 11 (4): 648–655, doi:10.1137/0911038, ISSN 0196-5204 [1]
Stewart, G. W. (1976), "The Economical Storage of Plane Rotations", Numerische Mathematik, 25 (2): 137–138, doi:10.1007/BF01462266, ISSN 0029-599X
Stewart, G. W. (1980), "The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators", SIAM J. Numer. Anal., 17 (3): 403–409, doi:10.1137/0717034, ISSN 0036-1429
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Orthogonal matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Tutorial and Interactive Program on Orthogonal Matrix
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text:
Celtic Britons
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Great Britain and adjacent islands in the 5th century AD, before the invasion and subsequent founding of Anglo-Saxon kingdoms.
Mainly Brittonic areas
Mainly Pictish areas
Mainly Goidelic areas
The Britons, also known as Celtic Britons or Ancient Britons, were Celtic people who inhabited Great Britain from the British Iron Age into the Middle Ages, at which point their culture and language diverged into the modern Welsh, Cornish and Bretons (among others). They spoke the Common Brittonic language, the ancestor to the modern Brittonic languages.[1]
The traditional view that the Celtic Britons originally migrated from the continent, mostly across the English Channel, with their languages, culture and genes in the Iron Age has been considerably undermined in recent decades by the contention of many scholars that Celtic languages had instead spread north along the Atlantic seaboard during the Bronze Age,[2] and the results of genetic studies, which show a large continuity between Iron Age and older British populations,[3] suggesting trans-cultural diffusion was also very important in the introduction of the Celtic languages.
The earliest evidence for the Britons and their language in historical sources dates to the Iron Age.[4] After the Roman conquest of Britain in the 1st century, a Romano-British culture emerged, and Latin and British Vulgar Latin coexisted with Brittonic.[5] During and after the Roman era, the Britons lived throughout Britain. Their relationship with the Picts, who lived north of the Firth of Forth, has been the subject of much discussion, though most scholars now accept that the Pictish language was related to Common Brittonic, rather than a separate Celtic language.[6]
With the beginning of Anglo-Saxon settlement and Gaelic Scots in the 5th century, the culture and language of the Britons fragmented and much of their territory was taken over by the Anglo-Saxons and Scots Gaels. The extent to which this cultural and linguistic change was accompanied by wholesale changes in the population is still a matter of discussion. During this period some Britons migrated to mainland Europe and established significant settlements in Brittany (now part of France) as well as Britonia in modern Galicia, Spain.[4] By the 11th century, remaining Brittonic Celtic-speaking populations had split into distinct groups: the Welsh in Wales, the Cornish in Cornwall, the Bretons in Brittany, and the people of the Hen Ogledd ("Old North") in southern Scotland and northern England. Common Brittonic developed into the distinct Brittonic languages: Welsh, Cumbric, Cornish and Breton.[4]
Contents
[hide]
1 Name
2 Language
3 Archaeology and art
4 Territory
5 See also
6 Notes
7 References
8 External links
Name[edit]
Main article: Britain (placename)
Further information: Brittia
Gritstone bas-relief of Romano-British woman
The earliest known reference to the inhabitants of Britain seems to come from 4th century BC records of the voyage of Pytheas, a Greek geographer who made a voyage of exploration around the British Isles between 330 and 320 BC. Although none of his own writings remain, writers during the time of the Roman Empire made much reference to them. Pytheas called the islands collectively αἱ Βρεττανίαι (hai Brettaniai), which has been translated as the Brittanic Isles; he also used the term Pretannike. The peoples of these islands were called the Πρεττανοί (Prettanoi), Priteni, Pritani or Pretani. The group included Ireland, which was referred to as Ierne (Insula sacra "sacred island" as the Greeks interpreted it) "inhabited by the race of Hiberni" (gens hibernorum), and Britain as insula Albionum, "island of the Albions".[7][8] The term Pritani may have reached Pytheas from the Gauls, who possibly used it as their term for the inhabitants of the islands.[8]
The Anglo-Saxon Chronicle, which was originally compiled by the orders of King Alfred the Great in approximately 890, and subsequently maintained and added to by generations of anonymous scribes until the middle of the 12th century, starts with this sentence: "The island Britain is 800 miles long, and 200 miles broad, and there are in the island five nations: English, Welsh (or British, including the Cornish), Scottish, Pictish, and Latin. The first inhabitants were the Britons, who came from Armenia, and first peopled Britain southward." ("Armenia" is possibly a mistaken transcription of Armorica, an area in northwestern Gaul including modern Brittany.)[9]
The Latin name in the early Roman Empire period was Britanni or Brittanni, following the Roman conquest in AD 43.[10]
The Welsh word Brython was introduced into English usage by John Rhys in 1884 as a term unambiguously referring to the P-Celtic speakers of Great Britain, to complement Goidel; hence the adjective Brythonic referring to the group of languages.[11] "Brittonic languages" is a more recent coinage (first attested 1923 according to the Oxford English Dictionary) intended to refer to the ancient Britons specifically.
In English, the terms "Briton" and British for many centuries originally denoted only the ancient Celtic Britons and their descendants, most particularly the Welsh, Cornish and Bretons, who were seen as heirs to the ancient British people.[12] After the Acts of Union 1707, the terms British and Briton gradually came to be applied to all inhabitants of the Kingdom of Great Britain, including the English, Scottish and some Northern Irish.[13]
Language[edit]
Main articles: Common Brittonic and Brittonic languages
The Staffordshire Moorlands Pan
A flying rowan tree, considered magical by the ancient Britons
Britons migrated westwards during the Anglo-Saxon settlement of Britain
The Britons spoke an Insular Celtic language known as Common Brittonic. Brittonic was spoken throughout the island of Britain (in modern terms, England, Wales and Scotland), as well as offshore islands such as the Isle of Man, Scilly Isles, Orkneys, Hebrides, Isle of Wight and Shetlands.[4][14] According to early medieval historical tradition, such as The Dream of Macsen Wledig, the post-Roman Celtic-speakers of Armorica were colonists from Britain, resulting in the Breton language, a language related to Welsh and identical to Cornish in the early period and still used today. Thus the area today is called Brittany (Br. Breizh, Fr. Bretagne, derived from Britannia).
Common Brittonic developed from the Insular branch of the Proto-Celtic language that developed in the British Isles after arriving from the continent in the 7th century BC. The language eventually began to diverge; some linguists have grouped subsequent developments as Western and Southwestern Brittonic languages. Western Brittonic developed into Welsh in Wales and the Cumbric language in the Hen Ogledd or "Old North" of Britain (modern northern England and southern Scotland), while the Southwestern dialect became Cornish in Cornwall and South West England and Breton in Armorica. Pictish is now generally accepted to descend from Common Brittonic, rather than being a separate Celtic language. Welsh and Breton survive today; Cumbric and Pictish became extinct in the 12th century. Cornish had become extinct by the 19th century but has been the subject of language revitalization since the 20th century.
Archaeology and art[edit]
Main article: Celtic art
Ideas about the development of British Iron Age culture changed greatly in the 20th century, and remain in development. Generally cultural exchange has tended to replace migration from the continent as the explanation for changes, although Aylesford-Swarling Pottery and the Arras culture of Yorkshire are examples of developments still thought to be linked to migration.
Although the La Tène style, which defines what is called Celtic art in the Iron Age, was late in arriving in Britain, after 300 BC the ancient British seem to have had generally similar cultural practices to the Celtic cultures nearest to them on the continent. There are significant differences in artistic styles, and the greatest period of what is known as the "Insular La Tène" style, surviving mostly in metalwork, was in the century or so before the Roman conquest, and perhaps the decades after it. By this time Celtic styles seem to have been in decline in continental Europe, even before Roman invasions.
An undercurrent of British influence is found in some artefacts from the Roman period, such as the Staffordshire Moorlands Pan, and it appears that it was from this, passing to Ireland in the late Roman and post-Roman period, that the "Celtic" element in Early Medieval Insular art derived.
Territory[edit]
Main articles: British Iron Age, Roman Britain, and Sub-Roman Britain
Throughout their existence, the territory inhabited by the Britons was composed of numerous ever-changing areas controlled by Brittonic tribes. The extent of their territory before and during the Roman period is unclear, but is generally believed to include the whole of the island of Great Britain, at least as far north as the Clyde-Forth isthmus, and if the Picts are included as Brittonic speaking people (as they more usually are),[15] the entirety of Great Britain and its offshore island groups. The territory north of the Firth of Forth was largely inhabited by the Picts; little direct evidence has been left of the Pictish language, but place names and Pictish personal names recorded in the later Irish annals suggest it was indeed related to the Common Brittonic language rather than to the Goidelic (Gaelic) languages of the Irish, Scots and Manx; indeed their Goidelic Irish name, Cruithne, is cognate with Brythonic Priteni. After the invasion of north western Britain by Gaelic speaking Celts from Ireland from the 6th century AD onwards,part of the Pictish territory was eventually absorbed into the Gaelic kingdoms of Dál Riata and Alba, which became Scotland. The Isle of Man, Shetland, Hebrides and the Orkney islands were originally inhabited by Britons also, but eventually became respectively Manx and Scots Gaelic speaking territories, while the Scilly isles and Anglesey (Ynys Mon) remained Brittonic and the originally Brittonic Isle of Wight was taken by Anglo-Saxons.
In 43 AD, the Roman Empire invaded Britain. The British tribes opposed the Roman legions for many decades, but by 84 AD the Romans had decisively conquered southern Britain and had pushed into Brittonic areas of what would later become northern England and southern Scotland. In 122 AD, they fortified the northern border with Hadrian's Wall, which spanned what is now Northern England. In 142 AD, Roman forces pushed north again and began construction of the Antonine Wall, which ran between the Forth-Clyde isthmus, but they retreated back to Hadrian's Wall after only twenty years. Although the native Britons south of Hadrian's Wall mostly kept their land, they were subject to the Roman governors, whilst the Brittonic-Pictish Britons north of the wall remained fully independent and unconquered. The Roman Empire retained control of "Britannia" until its departure about AD 410, although some parts of Britain had already effectively shrugged off Roman rule decades earlier.
Thirty years or so after the time of the Roman departure, the Germanic-speaking Anglo-Saxons began a migration to the eastern coast of Britain, where they began to establish their own kingdoms, and the Gaelic speaking Scots migrating from Dál nAraidi (modern Northern Ireland), did the same on the west coast of Scotland and the Isle of Man.[16][17]
At the same time, some Britons established themselves in what is now called Brittany. There they set up their own small kingdoms and the Breton language developed there from Brittonic Insular Celtic rather than Gaulish or Frankish. A further Brittonic colony, Britonia, was also set up at this time in Gallaecia in northwestern Spain.
Many of the old Brittonic kingdoms began to disappear in the centuries after the Anglo-Saxon and Scottish Gaelic invasions; Parts of the regions of modern East Anglia, East Midlands, North East England, Argyll and South East England were the first to fall to the Germanic and Gaelic Scots invasions;
5th Century AD; The kingdom of Ceint (modern Kent) fell in 456 AD, Linnuis (which stood astride modern Lincolnshire and Nottinghamshire) was subsumed as early as 500 AD and became the English Kingdom of Lindsey.
6th Century AD; Rhegin (essentially modern Sussex and eastern Hampshire) was likely fully conquered by 510 AD, Ynys Weith (Isle of Wight) fell in 530 AD,[18]Caer Colun (essentially modern Essex) by 540 AD, The Gaels arrived on the north west coast of Britain from Ireland, dispossessed the native Britons and founded Dal Riata which encompassed modern Argyll, Skye and Iona between 500 and 560 AD. Deifr (Deira) which encompassed modern day Teesside, Wearside, Tyneside and Humberside fell to the Anglo-Saxons in 559 AD and Deira became an Anglo-Saxon kingdom after this point. Caer Went had officially disappeared by 575 AD becoming the Anglo-Saxon kingdom of East Anglia. Gwent was only partly conquered; its capital Caer Gloui (Gloucester) was taken by the Anglo-Saxons in 577 AD, handing Gloucestershire and Wiltshire to the invaders, while the westernmost part remained in Brittonic hands, and continued to exist in modern Wales.
7th Century AD; Caer Lundein encompassing London, St. Albans and parts of the Home Counties fell from Brittonic hands by 600 AD,[19] Bryneich which existed in modern Northumbria and County Durham with its capital of Din Guardi (modern Bamburgh) and which included Ynys Metcaut (Lindisfarne) had fallen by 605 AD becoming Anglo-Saxon Bernicia.[20] Caer Celemion (in modern Hampshire and Berkshire) had fallen by 610 AD.[21] Elmet, a large kingdom which covered much of modern Yorkshire, Lancashire and Cheshire and likely had its capital at modern Leeds, was conquered by the Anglo-Saxons in 627 AD. Pengwern, which covered Staffordshire, Shropshire, Herefordshire and Worcestershire, was largely destroyed in 656 AD, with only its westernmost parts in modern Wales remaining under the control of the Britons,[22] and it is likely that Cynwidion which had stretched from modern Bedfordshire to Northamptonshire, fell in the same general period as Pengwern, though a sub-kingdom of Calchwynedd may have clung on in the Chilterns for a time.
8th Century AD; Novant which occupied Galloway and Carrick was soon subsumed by fellow Brittonic-Pictish polities by 700 AD. Aeron which encompassed modern Ayrshire[23] was conquered into the Anglo-Saxon kingdom of Northumbria by 700 AD.
Some Brittonic kingdoms, such as Rheged, which at its height encompassed much what is today Strathclyde, Cumbria, Northumberland, the Scottish borders, Lancashire and modern Greater Manchester and had a capital at Cair Ligualid (Carlisle), were able to successfully resist these incursions for some time, before the eastern part peacefully joined with the Anglo-Saxon kingdom of Bernicia-Northumberland by 730 AD, and the west was taken over by the fellow Britons of Ystrad Clud.[24][25] Similarly, the kingdom of Gododdin, which appears to have had its capital at Din Eidyn (modern Edinburgh and encompassed parts of modern Northumbria, County Durham, Lothian and Clackmannanshire endured until approximately 775 AD before being divided by fellow Brittonic Picts, Gaelic Scots and Anglo-Saxons.
9th Century AD; The Kingdom of Cait, covering modern Caithness, Sutherland, Orkneys and Shetlands was conquered by Gaelic Ccots in 871 AD. Dumnonia (encompassing Cornwall, Devonshire and the Scilly Isles was partly conquered during the mid 9th century AD, with most of modern Devonshire being annexed by the Anglo-Saxons, but leaving Cornwall and the Scilly Isles still in the hands of the Britons, where they became Kernow.
10th Century AD; The Kingdom of Ce which encompassed modern Marr, Banff, Buchan, Fife and much of Aberdeenshire disappeared in 900 AD. Fortriu the largest Pictish kingdom which covered Strathearn, Morayshire and Easter Ross had fallen by approximately 950 AD to the Gaelic Kingdom of Alba (Scotland). Other Pictish kingdoms such as Circinn ( in modern Angus and The Mearns), Fib (modern Fife), Fidach (Inverness and Perthshire), Ath-Fotla (Atholl) had also fallen by the beginning of the 11th century AD.
The Brythonic languages in these areas was replaced by the Old English of the Anglo-Saxons and Scots Gaelic, although this was likely a gradual process in many areas.
Similarly, the Brittonic colony of Britonia in north western Spain appears to have disappeared soon after 900 AD.
11th Century AD; The kingdom of Ystrad Clud (Strathclyde) was for some time a large and powerful Brittonic kingdom of the Henn Ogledd (the 'Old North) which endured until the end of the 11th century, successfully resisting Anglo-Saxon, Gaelic Scots and later also Viking attacks. At its peak it encompassed modern Strathclyde, Dumbartonshire, Cumbria, Stirlingshire, Lanarkshire, Ayrshire, Dumfries and Galloway, Argyll and Bute and parts of North Yorkshire, the western Pennines, and as far as modern Leeds in West Yorkshire.[25][26] Thus the Kingdom of Strathclyde (Ystrad-Clud) became the last of the Brittonic kingdoms of the old north to fall in the 1090s, when it was effectively divided between England and Scotland.[27]
The Britons also retained control of Wales, Kernow (encompassing Cornwall and the Scilly Isles)) until the mid 11th century AD when Cornwall was annexed by the English, with the Isles of Scilly following a few years later.
Wales remained free from Anglo-Saxon, Gaelic Scots and Viking control, and was divided among varying Brittonic kingdoms, the foremost being Gwynedd, Powys (including Clwyd and Ynys Mon (Anglesey), Deheubarth (originally Ceredigion, Seisyllwg and Dyfed), Gwent and Morgannwg (Glamorgan. Some of these Welsh kingdoms initially included territories further east, for example Powys included parts of modern Merseyside, Cheshire and The Wirral and Gwent held parts of modern Herefordshire, Worcestershire, Somerset and Gloucestershire, but had largely been confined to the borders of modern Wales by the beginning of the 12th century.
However, by the beginning of the 12th century, the Anglo-Saxons and Gaels had become the dominant cultural force in most of the formerly Brittonic ruled territory in Britain, and the language and culture of the native Britons was thereafter gradually replaced in those regions,[28] remaining only in Wales, Cornwall, Scilly Isles and Brittany, and for a time in parts of Cumbria, Strathclyde, and eastern Galloway.
Cornwall (Kernow, Dumnonia) had certainly been largely absorbed by England by the 1050s, although it retained a distinct Brittonic culture and language.[29] Britonia in Spanish Galicia seems to have disappeared by 900 AD.
Wales and Brittany remained independent for some time however, with Brittany finally being absorbed into France during the 1490s, and Wales united with England by the Laws in Wales Acts 1535–1542 in the mid 16th century during the rule of the Tudors (Twdyr), who were themselves of Welsh heritage on the male side.
Wales, Cornwall and Brittany continued to retain a distinct Brittonic culture, identity and language, which they have maintained to the present day. The Welsh language and Breton language remain widely spoken, and the Cornish language, once close to extinction, has experienced a revival since the 20th century. The vast majority of place names and names of geographical features in Wales, Cornwall and Brittany are Brittonic, and Brittonic family and personal names remain common.
During the 19th century, a large number of Welsh farmers migrated to Patagonia in Argentina, forming a community called Y Wladfa, which today consists of over 1,500 Welsh speakers.
In addition, a Brittonic legacy remains in England, Scotland and Galicia in Spain,[30] in the form of often large numbers of Brittonic place and geographical names. Some examples of geographical Brittonic names survive in the names of rivers, such as the Thames, Clyde, Severn, Tyne, Wye, Exe, Dee, Tamar, Tweed, Avon, Trent, Tambre, Navia and River Forth. A number of place names in England and Scotland are of Brittonic rather than Anglo-Saxon or Gaelic origin, such as; London, Manchester, Glasgow, Edinburgh, Carlisle, Caithness, Aberdeen, Dundee, Barrow, Exeter, Lincoln, Dumbarton, Brent, Penge, Colchester, Durham, Dover, Leatherhead and York.
See also[edit]
Albion
British Latin
Celtic nations
Cornish people
Cumbric
English people
Fortriu
Genetic history of the British Isles
Gododdin
History of the British Isles
Kingdom of Cat
Kingdom of Ce
Kingdom of Strathclyde
List of Celtic tribes
Scottish people
Welsh people
Yr Hen Ogledd
Notes[edit]
Jump up ^ Koch, p. 291.
Jump up ^ Review by Joseph F. Eska in Bryn Mawr Classical Review (2013.12.35) of John T. Koch, Barry W. Cunliffe (eds.), Celtic from the West 2: Rethinking the Bronze Age and the Arrival of Indo-European in Atlantic Europe. Celtic studies publications, 16. Oxford; Oakville, CT: Oxbow Books, 2013. ISBN 9781842175293.
Jump up ^ Cristian Capelli; Nicola Redhead; Julia K. Abernethy; Fiona Gratrix; James F. Wilson; Torolf Moen; Tor Hervig; Martin Richards; Michael P. H. Stumpf; Peter A. Underhill; Paul Bradshaw; Alom Shaha; Mark G. Thomas; Neal Bradman & David B. Goldstein (2003). "A Y chromosome census of the British Isles". Current Biology. 13 (11): 979–984. doi:10.1016/S0960-9822(03)00373-7. PMID 12781138. ; McEvoy; Richards, M; Forster, P; Bradley, DG (2004). "The Longue Durée of Genetic Ancestry: Multiple Genetic Marker Systems and Celtic Origins on the Atlantic Facade of Europe". American Journal of Human Genetics. 75 (4): 693–702. doi:10.1086/424697. PMC 1182057  . PMID 15309688.
^ Jump up to: a b c d Koch, pp. 291–292.
Jump up ^ Sawyer, P.H. (1998). From Roman Britain to Norman England. pp. 69–74. ISBN 0415178940.
Jump up ^ Forsyth, p. 9.
Jump up ^ Snyder, Christopher A. (2003). The Britons. Blackwell Publishing. ISBN 0-631-22260-X.
^ Jump up to: a b Foster (editor), R F; Donnchadh O Corrain, Professor of Irish History at University College Cork: Prehistoric and Early Christian Ireland (1 November 2001). The Oxford History of Ireland. Oxford University Press. ISBN 0-19-280202-X. CS1 maint: Extra text: authors list (link)
Jump up ^ "The Avalon Project". Yale Law School. Retrieved 10 August 2011.
Jump up ^ OED s.v. "Briton". See also Online Etymology Dictionary: Briton
Jump up ^ Online Etymology Dictionary: Brythonic
Jump up ^ Roberts, Peter (2003). Bradshaw, Brendon; Roberts, Peter, eds. "Tudor Wales, national identity, and British inheritance". British Consciousness and Identity: the Making of Britain, 1533-1707. Cambridge University Press: 8. ISBN 9780521893619. Retrieved May 18, 2017.
Jump up ^ "Briton". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library membership required.)
Jump up ^ While there have been attempts in the past to align the Pictish language with non-Celtic language, the current academic view is that it was Brittonic. See: Forsyth (1997) p. 37: "[T]he only acceptable conclusion is that, from the time of our earliest historical sources, there was only one language spoken in Pictland, the most northerly reflex of Brittonic."
Jump up ^ Forsyth 2006, p. 1447; Forsyth 1997; Fraser 2009, pp. 52–53; Woolf 2007, pp. 322–340
Jump up ^ John E Pattison. Is it necessary to assume an apartheid-like social structure in early Anglo-Saxon England? Proceedings of the Royal Society B, 275(1650), 2423–2429, 2008 doi:10.1098/rspb.2008.0352
Jump up ^ Pattison, John E. (2011) "Integration versus Apartheid in post-Roman Britain: a Response to Thomas et al. (2008)," Human Biology: Vol. 83: Iss. 6, Article 9. pp. 715–733, 2011. Abstract available at: http://digitalcommons.wayne.edu/humbiol/vol83/iss6/9
Jump up ^ http://www.historyfiles.co.uk/FeaturesBritain/BritishSouthernBritain05.htm#Ynys Weith
Jump up ^ http://www.historyfiles.co.uk/FeaturesBritain/BritishSouthernBritain04.htm#Caer Lundein
Jump up ^ Koch, John T. (2006). Celtic Culture: a historical encyclopedia. ABC-CLIO. pp. 515–516.
Jump up ^ http://www.historyfiles.co.uk/KingListsBritain/BritainAtrebates.htm#Caer Celemion
Jump up ^ http://www.historyfiles.co.uk/FeaturesBritain/BritishSouthernBritain01.htm#Pengwern
Jump up ^ a b c d Bromwich, p. 157.
Jump up ^ Chadwick, H.M.; Chadwick, N.K. (1940). The Growth of Literature. 1. Cambridge: Cambridge University Press.
^ Jump up to: a b Kapelle, W.E. (1979). The Norman Conquest of the North: the Region and its Transformation, 1000–1135. Chapel Hill, NC: University of North Carolina Press. ISBN 0-7099-0040-6.
Jump up ^ Broun, "Dunkeld", Broun, "National Identity", Forsyth, "Scotland to 1100", pp. 28–32, Woolf, "Constantine II"; cf. Bannerman, "Scottish Takeover", passim, representing the "traditional" view.
Jump up ^ Charles-Edards, pp. 12, 575; Clarkson, pp. 12, 63-66, 154-58
Jump up ^ Germanic invaders may not have ruled by apartheid New Scientist, 23 April 2008
Jump up ^ Williams, Ann and Martin, G. H. (tr.) (2002) Domesday Book: a complete translation, London: Penguin, pp. 341–357.
Jump up ^ Young, Simon (2002). Britonia: camiños novos. Noia: Toxosoutos. pp. 123–128. ISBN 978-84-95622-58-7.
References[edit]
Forsyth, Katherine (1997). Language in Pictland (PDF). De Keltische Draak. ISBN 90-802785-5-6.
Koch, John T. (2006). Celtic Culture: A Historical Encyclopedia. ABC-CLIO.
The Anglo-Saxon Chronicle.
External links[edit]
BBC – History – Native Tribes of Britain
DNA from ethnic Britons found in Ireland
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