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The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoted aij. The table below uses the Kronecker delta δij for two integers i and j which is 1 if i = j and 0 else. Name Explanation Symbolic description of the entries Notes Exchange matrix A binary matrix with ones on the anti-diagonal, and zeroes everywhere else. aij = δn + 1 − i,j A permutation matrix. Hilbert matrix aij = (i + j − 1)−1. A Hankel matrix. Identity matrix A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0 aij = δij Lehmer matrix aij = min(i,j) ÷ max(i,j). A positive symmetric matrix. Matrix of ones A matrix with all entries equal to one aij = 1. Pascal matrix A matrix containing the entries of Pascal's triangle. Pauli matrices A set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I2 identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices. Redheffer matrix aij are 1 if i divides j or if j = 1; otherwise, aij = 0. A (0, 1)-matrix. Shift matrix A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. aij = δi+1,j or aij = δi−1,j Multiplication by it shifts matrix elements by one position. Zero matrix A matrix with all entries equal to zero. aij = 0. Matrices with conditions on eigenvalues or eigenvectors[edit] Name Explanation Notes Companion matrix A matrix whose eigenvalues are equal to the roots of the polynomial. Convergent matrix A square matrix whose successive powers approach the zero matrix. Its eigenvalues have magnitude less than one. Defective matrix A square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalisable. Diagonalizable matrix A square matrix similar to a diagonal matrix. It has an eigenbasis, that is, a complete set of linearly independent eigenvectors. Hurwitz matrix A matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix. Positive-definite matrix A Hermitian matrix with every eigenvalue positive. Stability matrix Synonym for Hurwitz matrix. Stieltjes matrix A real symmetric positive definite matrix with nonpositive off-diagonal entries. Special case of an M-matrix. |
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