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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]
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 ...
The program structure of this algorithm is a simple triple-loop, as in the standard Gaussian elimination. However in this case the matrix is modified so that each M k,k entry contains the leading principal minor [M] k,k. Algorithm correctness is easily shown by induction on k. [4]
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:
For a square matrix, the diagonal (or main diagonal or principal diagonal) is the diagonal line of entries running from the top-left corner to the bottom-right corner. [ 1 ] [ 2 ] [ 3 ] For a matrix A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle j} , these would be entries A i ...
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 .
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 ...
A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an integral operator.