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 aijand the vector v has elements
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.
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
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.
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
An ill-conditioned matrix is one where the solution to Ax=b is overly
sensitive to perturbations in A or b. See condition
A matrix is involutary if A2 = I.
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
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.
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.
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.