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Minor (linear algebra) In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down 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 in turn are useful for ...
A non-vanishing p-minor (p × p submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the converse is ...
Sylvester's criterion. 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: the upper left 1-by-1 ...
Totally positive matrix. In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. [1] A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive ...
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector , where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. [1][2][3][4] Compound matrices are closely related to exterior algebras, [5] and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying ...
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.
LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. [1]