Question: What is the explanation for a satisfying conditions on products or inverses complex square matrix A is normal if it commutes with its conjugate transpose A* ?

text: List of matrices
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 ( C ) i , j = ∑ r = 1 n A i , r B r , j . { \displaystyle ( C ) _ { i , j } =\sum _ { r=1 } ^ { n } A_ { i , r } B_ { r , j } . }
This page lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. An important example is the identity matrix given by
Name Explanation Notes
Circular matrix or Coninvolutory matrix A matrix whose inverse is equal to its entrywise complex conjugate : A = A Compare with unitary matrices
Congruent matrix Two matrices A and B are congruent if there exists an invertible matrix P such that P A P = B Compare with similar matrices
EP matrix or Range-Hermitian matrix A square matrix that commutes with its Moore-Penrose inverse : AA = A A
Idempotent matrix or Projection Matrix A matrix that has the property A ² = AA = A The name projection matrix inspires from the observation of projection of a point multiple times onto a subspace ( plane or a line ) giving the same result as one projection
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
Involutory matrix A square matrix which is its own inverse , i.e. , AA = I Signature matrices , Householder Matrices ( Also known as 'reflection matrices ' to reflect a point about a plane or line ) have this property
Nilpotent matrix A square matrix satisfying A = 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 = A 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
Unitary matrix A square matrix whose inverse is equal to its conjugate transpose , A = A
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 AA = wI for some positive integer w