- Laplace expansion
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This article is about the expansion of the determinant of a square matrix as a weighted sum of determinants of sub-matrices. For the expansion of an 1/r-potential using spherical harmonical functions, see Laplace expansion (potential).
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 i, j ∈ {1, 2, ..., n}.
Then its determinant |B| is given by:
Contents
Examples
Consider the matrix
The determinant of this matrix can be computed by using the Laplace expansion along the first row:
Alternatively, Laplace expansion along the second column yields
It is easy to see that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.
Proof
Suppose B is an n × n matrix and
For clarity we also label the entries of B that compose its i,j minor matrix Mij as
(ast) for
Consider the terms in the expansion of | B | that have bij as a factor. Each has the form
for some permutation τ ∈ Sn with τ(i) = j, and a unique and evidently related permutation
which selects the same minor entries as τ. Similarly each choice of σ determines a corresponding τ, i.e. the correspondence
is a bijection between Sn − 1 and
The permutation τ can be derived from σ as follows.
Define
by σ'(k) = σ(k) for
and σ'(n) = n. Then sgn σ' = sgn σ and
Since the two cycles can be written respectively as n − j and n − i transpositions,
And since the map
is bijective,
from which the result follows.
References
- David Poole: Linear Algebra. A Modern Introduction. Cengage Learning 2005, ISBN 0534998453, p. 265-267 (restricted online copy at Google Books)
- Harvey E. Rose: Linear Algebra. A Pure Mathematical Approach. Springer 2002, ISBN 3764369051, p. 57-60 (restricted online copy at Google Books)
See also
External links
Categories:- Matrix theory
- Determinants
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