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A is upper bidiagonal if a(i,j)=0 unless i=j or i=j-1.
A is lower bidiagonal if a(i,j)=0 unless i=j or i=j+1
A bidiagonal matrix is also tridiagonal, triangular, , and Hessenberg.
A is block diagonal if it has the form [A 0 ... 0; 0 B ... 0;...;0 0 ... Z] where A, B, ..., Z are matrices (not necessarily square).
A circulant matrix, A, is an n*n Toeplitz matrix in which a(i,j) is a function of {(i-j) modulo n}. In other words each column of A is equal to the previous column rotated downwards by one element.
A Circular matrix, A, is one for which inv(A) = conj(A).
If p(x) is a polynomial of the form a(0) + a(1)*x + a(2)*x2 + ... + a(n)*xn then the polynomial's companion matrix is n#n and equals [0 I; -a(0:n-1)/a(n)] where I is n-1#n-1.
The rows and columns are sometimes given in reverse order [-a(n-1:0)/a(n) ; I 0].
A matrix A is convergent if Ak tends to 0 as k tends to infinity.
WARNING: The term reducible is sometimes used instead of decomposable.
A matrix, A, is decomposable if there exists a permutation matrix P
such that PTAP is of the form [B 0; C D]
where B and D are square.
A matrix that is not decomposable is indecomposable.
A matrix, A, is partly-decomposable if there exist permutation matrices P
and Q such that PTAQ is of the form [B 0; C
D] where B and D are square.
A matrix that is not even partly-decomposable is fully-indecomposable.
A matrix A:n#n is defective if it does not have n linearly independent eigenvectors, otherwise it is simple.
A real or Hermitian square matrix A is positive
definite if xHAx > 0 for all non-zero x.
A real or Hermitian square matrix A is positive semi-definite or non-negative
definite if xHAx >=0 for all non-zero x.
A real or Hermitian square matrix A is indefinite
if xHAx is > 0 for some x and < 0 for
some other x.
Note that for any non-Hermitian complex matrix xHAx is complex for some values of x. Such matrices are therefore excluded from the concept of definiteness.
An n*n square matrix is derogatory if its minimal polynomial is of lower order than n.
A is diagonal if a(i,j)=0 unless i=j.
A matrix is simple or diagonable if it is similar to a diagonal matrix otherwise it is defective.
A square matrix A is diagonally dominant if the absolute value of each diagonal element is greater than the sum of absolute values of the non-diagonal elements in its row. That is if for all i |a(i,i)| > Sum(|a(i,j)|;j != i).
A real non-negative square matrix A is doubly-stochastic if its rows and columns all sum to 1.
See under stochastic for properties.
The exchange matrix J:n*n is equal to [e(n) e(n-1) ... e(2) e(1)]. It is equal to I but with the columns in reverse order.
[Real]: A Givens Reflection is an n*n matrix of the form PT[Q 0 ; 0 I]P where P is any permutation matrix and Q is a matrix of the form [cos(x) sin(x); sin(x) -cos(x)].
[Real]: A Givens Rotation is an n*n matrix of the form PT[Q 0 ; 0 I]P where P is a permutation matrix and Q is a matrix of the form [cos(x) sin(x); -sin(x) cos(x)].
An n*n Hadamard matrix has orthogonal columns whose elements are all equal to +1 or -1.
A real 2n*2n matrix, A, is Hamiltonian if KA is symmetric where K = [0 I; -I 0].
See also: symplectic
A Hankel matrix has constant anti-diagonals. In other words a(i,j) depends only on (i+j).
A square matrix A is Hermitian if A = AH, that is A(i,j)=conj(A(j,i))
For real matrices, Hermitian and symmetric are equivalent. Except where stated, the following properties apply to real symmetric matrices as well.
A Hessenberg matrix is like a triangular matrix except that the elements
adjacent to the main diagonal can be non-zero.
A is upper Hessenberg if A(i,j)=0 whenever i>j+1. It is like
an upper triangular matrix except for the elements immediately
below the main diagonal.
A is lower Hessenberg if a(i,j)=0 whenever i<j-1. It is
like a lower triangular matrix except for the elements
immediately above the main diagonal.
A Hilbert matrix is a square Hankel matrix with elements a(i,j)=1/(i+j-1).
A Householder matrix (also called Householder reflection or transformation) is a matrix of the form (I-2vvH) for some vector v with ||v||=1.
Multiplying a vector by a Householder transformation reflects it in the hyperplane that is orthogonal to v.
Householder matrices are important because they can be chosen to annihilate any contiguous block of elements in any chosen vector.
The hypercompanion matrix of the polynomial p(x)=(x-a)n is aI:n#n+[0 I:n-1#n-1; 0]. It is an upper bidiagonal matrix that is zero except for the value a along the main diagonal and the value 1 on the diagonal immediately above it.
P matrix P is idempotent if P2 = P . An idempotent matrix that is also hermitian is called a projection matrix.
WARNING: Some people call any idempotent matrix a projection matrix and call it an orthogonal projection matrix if it is also hermitian.
If P is idempotent:
The identity matrix , I, has a(i,i)=1 for all i and a(i,j)=0 for all i !=j
An incidence matrix is one whose elements all equal 1 or 0.
An Integral matrix is one whose elements are all integers.
An Involutary matrix is one whose square equals the identity.
see under Tridiagonal
A matrix A is nilpotent to index k if Ak = 0 but Ak-1 != 0.
see under positive
A square matrix A is normal if AHA = AAH
A real square matrix A is orthogonal if ATA = I
Geometrically: Orthogonal matrices correspond to rotations and reflections.
Most properties are listed under unitary.
A square matrix A is a permutation matrix if its columns are a permutation of the columns of I.
An n*n matrix A is persymmetric if it is symmetric about is anti-diagonal, i.e. if a(i,j) = a(n+1-j,n+1-i).
A polynomial matrix of order p is one whose elements are polynomials of a single variable x. Thus A=A(0)+A(1)x+...+A(p)xp where the A(i) are constant matrices and A(p) is not all zero.
See also regular.
A real matrix is positive if all its elements are strictly > 0.
A real matrix is non-negative if all its elements are >= 0.
see under definiteness
If k is the eigenvalue of a matrix A having the largest absolute value, then A is primitive if the absolute values of all other eigenvalues are < |k|.
A projection matrix (or orthogonal projection matrix) is a square matrix that is hermitian and idempotent: i.e. PH=P2=P.
WARNING: Some people call any idempotent matrix a projection matrix and call it an orthogonal projection matrix if it is also hermitian.
WARNING: The term reducible is sometimes used to mean decomposable.
A matrix A:n*n is reducible if it is similar to a block-diagonal matrix of the form [B 0; 0 C] where B and C are square.
A polynomial matrix, A, of order p is regular if det(A) is non-zero.
An n*n square matrix is simple or diagonable if it has n linearly independent eigenvectors, otherwise it is defective.
A matrix is singular if it has no inverse.
A square matrix A is Skew-Hermitian if A = -AH, that is a(i,j)=-conj(a(j,i))
For real matrices, Skew-Hermitian and skew-symmetric are equivalent. The following properties apply also to real skew-symmetric matrices.
A square matrix A is skew-symmetric if A = -AT, that is a(i,j)=-a(j,i)
For real matrices, skew-symmetric and Skew-Hermitian are equivalent. Most properties are listed under skew-Hermitian .
A matrix is sparse if it has relatively few non-zero elements.
A Stability or Stable matrix is one whose eigenvalues all have
strictly negative real parts.
A semi-stable matrix is one whose eigenvalues all have non-positive real parts.
A real non-negative square matrix A is stochastic
if all its rows sum to 1.
See also Doubly Stochastic
A real non-negative square matrix A is sub-stochastic if all its rows sum to <=1.
A is subunitary if ||AAHx|| = ||AHx|| for all x. A is also called a partial isometry.
The following are equivalent:
A square matrix A is symmetric if A = AT, that is a(i,j) = a(j,i).
Most properties of real symmetric matrices are listed under Hermitian .
See also Hankel.
A real matrix, A, is symmetrizable if ATM = MA for some positive definite M.
A real matrix, A[2n#2n], is symplectic if ATKA=K is symmetric where K = [0 I; -I 0].
See also: hamiltonian
A toeplitz matrix has constant diagonals. In other words a(i,j) depends only on (i-j).
A is upper triangular if a(i,j)=0 whenever i>j.
A is lower triangular if a(i,j)=0 whenever i<j.
A is triangular iff it is either upper or lower triangular.
A triangular matrix A is strictly triangular if its diagonal elements
all equal 0.
A triangular matrix A is unit triangular if its diagonal elements all
equal 1.
A is tridiagonal if A(i,j)=0 whenever |i-j|>1. In other words its non-zero elements lie either on or immediately adjacent to the main diagonal.
A complex square matrix A is unitary if AHA = I. A is also sometimes called an isometry.
A real unitary matrix is called orthogonal .The following properties apply to orthogonal matrices as well as to unitary matrices.
An n*n Vandermonde matrix is of the form [xn-1 ... x2 x 1] for some column vector x. a general element is given by v(i,j)=x(i)n-j. The rightmost column of the matrix has all elements = 1.
Vandermonde matrices arise in connection with fitting polynomials to data.
The zero matrix, 0, has a(i,j)=0 for all i,j
The Matrix Reference Manual is written by Mike Brookes, Imperial College, London, UK. Please send any comments or suggestions to mike.brookes@ic.ac.uk