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A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
A matrix with relatively few non-zero elements. Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms. Symmetric matrix: A square matrix which is equal to its transpose, A = A T (a i,j = a j,i). Toeplitz matrix: A matrix with constant diagonals. Totally positive matrix
The definition of positive definite can be generalized by designating any complex matrix (e.g. real non-symmetric) as positive definite if {} > for all non-zero complex vectors , where {} denotes the real part of a complex number . [19] Only the Hermitian part (+) determines whether the matrix is positive definite, and is assessed in the ...
If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfies A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex ...
Invertible matrices are analogous to non-zero complex numbers. The inverse of a matrix has each eigenvalue inverted. A uniform scaling matrix is analogous to a constant number. In particular, the zero is analogous to 0, and; the identity matrix is analogous to 1. An idempotent matrix is an orthogonal projection with each eigenvalue either 0 or 1.
The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial ...
In mathematics, persymmetric matrix may refer to: a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or; a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line. The first definition is the most common in the recent literature.
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × ...