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In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
If the matrix is symmetric indefinite, it may be still decomposed as = where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, and is a direct sum of symmetric and blocks, which is called Bunch–Kaufman decomposition [6]
The derivation of the result hinges on a few basic observations: The real matrix / /, with (), is well-defined and skew-symmetric.; Any skew-symmetric real matrix can be block-diagonalized via orthogonal real matrices, meaning there is () such that = with a real positive-definite diagonal matrix containing the singular values of .
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
The particular system of equations was used by Morris to illustrate the concept of ill-conditioned system of equations. The matrix has been used as an example and for test purposes in many research papers and books over the years. John Todd has referred to as “the notorious matrix W of T. S. Wilson”. [1]
A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for n > 2. From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts. As it is a Z ...
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (as matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main ...
Let X be a p × p symmetric matrix of random variables that is positive semi-definite. Let V be a (fixed) symmetric positive definite matrix of size p × p. Then, if n ≥ p, X has a Wishart distribution with n degrees of freedom if it has the probability density function