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We use uppercase letters to represent matrices, lowercase letters to represent vectors, and subscripted lowercase letters to represent matrix or vector elements. Thus, the matrix A has elements denoted aij and the vector v has elements vj.

A banded matrix has its non-zero elements within a `band' about the diagonal. The bandwidth of a matrix A is defined as the maximum of |i-j| for which aij is nonzero. The upper bandwidth is the maximum j-i for which aij is nonzero and j>i. See diagonal, tridiagonal and triangular matrices as particular cases.

Condition number
The condition number of a matrix A is the quantity ||A||2 ||A-1||2. It is a measure of the sensitivity of the solution of Ax=b to perturbations of A or b. If the condition number of A is `large', A is said to be ill-conditioned. If the condition number is one, A is said to be perfectly conditioned. (The Matrix Market provides condition number estimates based on Matlab's condest() function which uses Higham's modification of Hager's one-norm method.)

A defective matrix has at least one defective eigenvalue, i.e. one whose algebraic multiplicity is greater than its geometric multiplicity. A defective matrix cannot be transformed to a diagonal matrix using similarity transformations.

A matrix A is positive definite if xT A x > 0 for all nonzero x. Positive definite matrices have other interesting properties such as being nonsingular, having its largest element on the diagonal, and having all positive diagonal elements. Like diagonal dominance, positive definiteness obviates the need for pivoting in Gaussian elimination. A positive semidefinite matrix has xT A x >= 0 for all nonzero x. Negative definite and negative semidefinite matrices have the inequality signs reveresed above.

A diagonal matrix has its only non-zero elements on the main diagonal.

Diagonal Dominance
A matrix is diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row (or column). Pivoting in Gaussian elimination is not necessary for a diagonally dominant matrix.

A matrix A is a Hankel matrix if the anti-diagonals are constant, that is, aij = fi+j for some vector f.

A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal). A nonsymmetric matrix can always be reduced to Hessenberg form by a finite sequence of similarity transformations.

A Hermitian matrix A is self adjoint, that is AH = A, where AH, the adjoint, is the complex conjugate of the transpose of A.

The Hilbert matrix A has elements aij = 1/(i+j-1). It is symmetric, positive definite, totally positive, and a Hankel matrix.

A matrix is idempotent if A2 = A.

Ill conditioned
An ill-conditioned matrix is one where the solution to Ax=b is overly sensitive to perturbations in A or b. See condition number.

A matrix is involutary if A2 = I.

Jordan block
The Jordan normal form of a matrix is a block diagonal form where the blocks are Jordan blocks. A Jordan block has its non-zeros on the diagonal and the first upper off diagonal. Any matrix may be transformed to Jordan normal form via a similarity transformation.

A matrix is an M-matrix if aij <= 0 for all i different from j and all the eigenvalues of A have nonnegative real part. Equivalently, a matrix is an M-matrix if aij <= 0 for all i different from j and all the elements of A-1 are nonnegative.

A matrix is nilpotent if there is some k such that Ak = 0.

A matrix is normal if A AH = AH A, where AH is the conjugate transpose of A. For real A this is equivalent to A AT = AT A. Note that a complex matrix is normal if and only if there is a unitary Q such that QH A Q is diagonal.

A matrix is orthogonal if AT A = I. The columns of such a matrix form an orthogonal basis.

The rank of a matrix is the maximum number of independent rows or columns. A matrix of order n is rank deficient if it has rank < n.

A singular matrix has no inverse. Singular matrices have zero determinants.

Symmetric/ Skew-symmetric
A symmetric matrix has the same elements above the diagonal as below it, that is, aij = aji, or A = AT. A skew-symmetric matrix has aij = -aji, or A = -AT; consequently, its diagonal elements are zero.

A matrix A is a Toeplitz if its diagonals are constant; that is, aij = fj-i for some vector f.

Totally Positive/Negative
A matrix is totally positive (or negative, or non-negative) if the determinant of every submatrix is positive (or negative, or non-negative).

An upper triangular matrix has its only non-zero elements on or above the main diagonal, that is aij=0 if i>j. Similarly, a lower triangular matrix has its non-zero elements on or below the diagonal, that is aij=0 if i<j.

A tridiagonal matrix has its only non-zero elements on the main diagonal or the off-diagonal immediately to either side of the diagonal. A symmetric matrix can always be reduced to a symmetric tridiagonal form by a finite sequence of similarity transformations.

A unitary matrix has AH = A-1.

The Matrix Market is a service of the Mathematical and Computational Sciences Division / Information Technology Laboratory / National Institute of Standards and Technology

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Last change in this page : December 3, 1999. [ ].