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Adjugate matrix. In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj (A). [1][2] It is also occasionally known as adjunct matrix, [3][4] or "adjoint", [5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate ...
Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
Laplace expansion. In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n - matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) - submatrices of B. Specifically, for every i, the Laplace expansion ...
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 computing both the determinant and inverse of square matrices.
The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors however, use the term transpose to refer to the adjoint as defined here. The adjoint allows us to consider whether g : Y → X is equal to u −1 : Y → X.
Laplacian matrix. In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace ...
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose —that is, the element in the i -th row and j -th column is equal to the complex conjugate of the element in the j -th row and i -th column, for all indices i and j: or in matrix form:
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This (n1 + ⋯ + nr) × (n1 + ⋯ + nr) square matrix, consisting of r diagonal blocks, can be compactly indicated as or , where the i -th Jordan block is Jλi,ni. For example, the matrix is a 10 × 10 Jordan matrix with a 3 × 3 block with eigenvalue 0, two 2 ...