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Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the ...
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:
A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices.
He did important research on invariant factors, integral matrices, principal submatrices, and the Baker-Campbell-Hausdorff formula. [7] [10] His research was honored with his appointment as lecturer for the 1988 Johns Hopkins Summer Lecture Series. [8]
The i-th Hurwitz determinant is the i-th leading principal minor (minor is a determinant) of the above Hurwitz matrix H. There are n Hurwitz determinants for a characteristic polynomial of degree n .
A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements. Boolean matrix: A matrix whose entries are taken from a Boolean algebra. Cauchy matrix: A matrix whose elements are of the form 1/(x i + y j) for (x i), (y j) injective sequences (i.e., taking every value only once). Centrosymmetric matrix
every principal submatrix of A is copositive as well. In particular, the entries on the main diagonal must be nonnegative. the spectral radius ρ(A) is an eigenvalue of A. [3] Every copositive matrix of order less than 5 can be expressed as the sum of a positive semidefinite matrix and a nonnegative matrix. [4]