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∗ = A∗A | 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, A − I 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. |
Read more about this topic: List Of Matrices
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