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Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix of order , the matrix in the -th stage of the computation (where the first stage = corresponds to the matrix itself) consists of all the connected minors of order of , where a connected minor is the determinant of a connected sub-block ...
For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo (the latter determinant ...
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
An invertible matrix with entries in the integers (integer matrix) Necessarily the determinant is +1 or −1. Unipotent matrix: A square matrix with all eigenvalues equal to 1. Equivalently, A − I is nilpotent. See also unipotent group. Unitary matrix: A square matrix whose inverse is equal to its conjugate transpose, A −1 = A *. Totally ...
For example, if is a square matrix representing a rotation (rotation matrix) and is a column ... The determinant of this matrix is −1, ...
The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant.
These relations are a direct consequence of the basic properties of determinants: evaluation of the (i, j) entry of the matrix product on the left gives the expansion by column j of the determinant of the matrix obtained from M by replacing column i by a copy of column j, which is det(M) if i = j and zero otherwise; the matrix product on the ...
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]