Special Matrices

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Bidiagonal

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.


Block Diagonal

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).


Circulant

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.


Circular

A Circular matrix, A, is one for which inv(A) = conj(A).


Companion Matrix

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].


Convergent

A matrix A is convergent if Ak tends to 0 as k tends to infinity.


Decomposable

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.


Defective[!]

A matrix A:n#n is defective if it does not have n linearly independent eigenvectors, otherwise it is simple.


Definiteness

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.


Derogatory

An n*n square matrix is derogatory if its minimal polynomial is of lower order than n.


Diagonal

A is diagonal if a(i,j)=0 unless i=j.


Diagonable or Simple or Non-Defective

A matrix is simple or diagonable if it is similar to a diagonal matrix otherwise it is defective.


Diagonally Dominant

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).


Doubly-Stochastic

A real non-negative square matrix A is doubly-stochastic if its rows and columns all sum to 1.

See under stochastic for properties.


Exchange

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.


Givens Reflection

[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)].


Givens Rotation

[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)].


Hadamard [!]

An n*n Hadamard matrix has orthogonal columns whose elements are all equal to +1 or -1.


Hamiltonian

A real 2n*2n matrix, A, is Hamiltonian if KA is symmetric where K = [0 I; -I 0].

See also: symplectic


Hankel

A Hankel matrix has constant anti-diagonals. In other words a(i,j) depends only on (i+j).


Hermitian

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.


Hessenberg

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.


Hilbert

A Hilbert matrix is a square Hankel matrix with elements a(i,j)=1/(i+j-1).


Householder

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.


Hypercompanion

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.


Idempotent [!]

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:


Identity[!]

The identity matrix , I, has a(i,i)=1 for all i and a(i,j)=0 for all i !=j


Incidence

An incidence matrix is one whose elements all equal 1 or 0.


Integral

An Integral matrix is one whose elements are all integers.


Involutary

An Involutary matrix is one whose square equals the identity.


Jacobi

see under Tridiagonal


Nilpotent [!]

A matrix A is nilpotent to index k if Ak = 0 but Ak-1 != 0.


Non-negative

see under positive


Normal

A square matrix A is normal if AHA = AAH


Orthogonal [!]

A real square matrix A is orthogonal if ATA = I

Geometrically: Orthogonal matrices correspond to rotations and reflections.

Most properties are listed under unitary.


Permutation

A square matrix A is a permutation matrix if its columns are a permutation of the columns of I.


Persymmetric

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).


Polynomial Matrix

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.


Positive

A real matrix is positive if all its elements are strictly > 0.
A real matrix is non-negative if all its elements are >= 0.


Positive Definite

see under definiteness


Primitive

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|.


Projection

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.


Reducible

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.


Regular

A polynomial matrix, A, of order p is regular if det(A) is non-zero.


Simple

An n*n square matrix is simple or diagonable if it has n linearly independent eigenvectors, otherwise it is defective.


Singular

A matrix is singular if it has no inverse.


Skew-Hermitian

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.


Skew-Symmetric[!]

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 .


Sparse

A matrix is sparse if it has relatively few non-zero elements.


Stability

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.


Stochastic

A real non-negative square matrix A is stochastic if all its rows sum to 1.!

See also Doubly Stochastic


Sub-stochastic

A real non-negative square matrix A is sub-stochastic if all its rows sum to <=1.


Subunitary

A is subunitary if ||AAHx|| = ||AHx|| for all x. A is also called a partial isometry.

The following are equivalent:

  1. A is subunitary
  2. AHA is a projection matrix
  3. AAHA = A
  4. A+ = AH

Symmetric[!]

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.


Symmetrizable

A real matrix, A, is symmetrizable if ATM = MA for some positive definite M.


Symplectic

A real  matrix, A[2n#2n], is symplectic if ATKA=K is symmetric where K = [0 I; -I 0].

See also: hamiltonian


Toeplitz

A toeplitz matrix has constant diagonals. In other words a(i,j) depends only on (i-j).


Triangular

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.


Tridiagonal or Jacobi

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.


Unitary

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.


Vandermonde

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.


Zero

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