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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.
A basic minor of a matrix is the determinant of a square submatrix that is of maximal size with nonzero determinant. [3] For Hermitian matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness. See Sylvester's criterion for more details.
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:
no LU factorization if the first (n−1) columns are non-zero and linearly independent and at least one leading principal minor is zero. In Case 3, one can approximate an LU factorization by changing a diagonal entry a j j {\displaystyle a_{jj}} to a j j ± ε {\displaystyle a_{jj}\pm \varepsilon } to avoid a zero leading principal minor.
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. [1] A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues).
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]
Now construct () householder matrix in a similar manner as such that maps the first column of ′ ′ to ‖ ′ ′ ‖, where ′ ′ is the submatrix of ′ constructed by removing the first row and the first column of ′, then let = [] which maps to the matrix which has only zeros below the first and second entry of the subdiagonal.