List of Matrices - Matrices Satisfying Conditions On Products or Inverses

Matrices Satisfying Conditions On Products or Inverses

A number of matrix-related notions is about properties of products or inverses of the given matrix. The matrix product of a m-by-n matrix A and a n-by-k matrix B is the m-by-k matrix C given by

This matrix product is denoted AB. Unlike the product of numbers, matrix products are not commutative, that is to say AB need not be equal to BA. A number of notions are concerned with the failure of this commutativity. An inverse of square matrix A is a matrix B (necessarily of the same dimension as A) such that AB = I. Equivalently, BA = I. An inverse need not exist. If it exists, B is uniquely determined, and is also called the inverse of A, denoted A−1.

Name Explanation Notes
Congruent matrix Two matrices A and B are congruent if there exists an invertible matrix P such that PT A P = B. Compare with similar matrices.
Idempotent matrix A matrix that has the property A² = AA = A.
Invertible matrix A square matrix having a multiplicative inverse, that is, a matrix B such that AB = BA = I. Invertible matrices form the general linear group.
Involutary matrix A square matrix which is its own inverse, i.e., AA = I. Signature matrices have this property.
Nilpotent matrix A square matrix satisfying Aq = 0 for some positive integer q. Equivalently, the only eigenvalue of A is 0.
Normal matrix A square matrix that commutes with its conjugate transpose: AA∗ = AA They are the matrices to which the spectral theorem applies.
Orthogonal matrix A matrix whose inverse is equal to its transpose, A−1 = AT. They form the orthogonal group.
Orthonormal matrix A matrix whose columns are orthonormal vectors.
Singular matrix A square matrix that is not invertible.
Unimodular matrix An invertible matrix with entries in the integers (integer matrix) Necessarily the determinant is +1 or −1.
Unipotent matrix A square matrix with all eigenvalues equal to 1. Equivalently, AI is nilpotent. See also unipotent group.
Totally unimodular matrix A matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program.
Weighing matrix A square matrix the entries of which are in {0, 1, −1}, such that AAT = wI for some positive integer w.

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