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In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices.
Formed by the cofactors of a square matrix, that is, the signed minors, of the matrix: Transpose of the Adjugate matrix: Companion matrix: A matrix having the coefficients of a polynomial as last column, and having the polynomial as its characteristic polynomial: Coxeter matrix
is called Hurwitz matrix corresponding to the polynomial . It was established by Adolf Hurwitz in 1895 that a real polynomial with a 0 > 0 {\displaystyle a_{0}>0} is stable (that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix H ( p ) {\displaystyle H(p)} are positive:
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
If is a singular matrix of rank , then it admits an LU factorization if the first leading principal minors are nonzero, although the converse is not true. [9] If a square, invertible matrix has an LDU (factorization with all diagonal entries of L and U equal to 1), then the factorization is unique. [8]
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In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries , where =. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones: