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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]
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]
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 ...
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:
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 ...
Compute the Sylvester matrix associated to and (). Rearrange each row in such a way that an odd row and the following one have the same number of leading zeros. Compute each principal minor of that matrix. If at least one of the minors is negative (or zero), then the polynomial f is not stable.
Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.