Determinant Expansion By Minors
Determinant Expansion by Minors
Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing thedeterminant of a given square matrix . Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large.
Let denote the determinant of a matrix , then
where is a so-called minor of , obtained by taking the determinant of with row and column "crossed out."
For example, for a matrix, the above formula gives
The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor is sometimes absorbed into the minor as
in which case is called a cofactor.
The equation for the determinant can also be formally written as
where ranges over all permutations of and is the inversion number of (Bressoud and Propp 1999).
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.
The i, j cofactor of B is the scalar Cij defined by
where Mij is the i, j minor matrix of B, that is, the determinant of the (n–1) × (n–1) matrix that results from deleting the i-th row and the j-th column of B.
Then the Laplace expansion is given by the following
Theorem. Suppose B = (bij) is an n × n matrix and fix any i, j ∈ {1, 2, ..., n}.
Then its determinant |B| is given by:
EXAMPLES
Consider the matrix
The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields:
Laplace expansion along the second column yields the same result:
It is easy to verify that
Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing thedeterminant of a given square matrix . Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large.
Let denote the determinant of a matrix , then
where is a so-called minor of , obtained by taking the determinant of with row and column "crossed out."
For example, for a matrix, the above formula gives
The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor is sometimes absorbed into the minor as
in which case is called a cofactor.
The equation for the determinant can also be formally written as
where ranges over all permutations of and is the inversion number of (Bressoud and Propp 1999).
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n square matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1). The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.
The i, j cofactor of B is the scalar Cij defined by
where Mij is the i, j minor matrix of B, that is, the determinant of the (n–1) × (n–1) matrix that results from deleting the i-th row and the j-th column of B.
Then the Laplace expansion is given by the following
Theorem. Suppose B = (bij) is an n × n matrix and fix any i, j ∈ {1, 2, ..., n}.
Then its determinant |B| is given by:
EXAMPLES
Consider the matrix
The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields:
Laplace expansion along the second column yields the same result:
It is easy to verify that