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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 ...
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 ...
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:
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:
In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P 0 {\displaystyle P_{0}} -matrices, which are the closure of the class of P -matrices, with every principal minor ≥ {\displaystyle \geq } 0.
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.