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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:
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 ...
If is invertible, then it admits an LU (or LDU) factorization if and only if all its leading principal minors [7] are nonzero [8] (for example [] does not admit an LU or LDU factorization). If A {\textstyle A} is a singular matrix of rank k {\textstyle k} , then it admits an LU factorization if the first k {\textstyle k} leading principal ...
In algebraic geometry, given a line bundle L on a smooth variety X, the bundle of n-th order principal parts of L is a vector bundle of rank (+ ()) that, roughly ...
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]
where Δ i is the i-th leading principal minor of the Hurwitz matrix associated with f. Using the same notation as above, the Liénard–Chipart criterion is that f is Hurwitz stable if and only if any one of the four conditions is satisfied:
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 .