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If the matrix that corresponds to a principal minor is a square upper-left submatrix of the larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to k, also known as a leading principal submatrix), then the principal minor is called a leading principal minor (of order k) or corner (principal) minor (of order k). [3]
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.
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]
Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first leading principal minors are neglected, the smallest minor consisting of the truncated first + rows and columns, the next consisting of the truncated first + rows ...
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 .
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.
German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. [2]