Determinant identities

From Wikipedia, the free encyclopedia

In mathematics the determinant is an operator which has certain useful identities.

Identities

where In is the n × n identity matrix.

For square matrices A and B of equal size,

for an n × n matrix.

If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

Schur complement

The following identity holds for a Schur complement of a square matrix:

The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with the "block lower triangular" matrix

Here Ip denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is

That is, we have effected a Gaussian decomposition

The first and last matrices on the RHS have determinant unity, so we have

This is Schur's determinant identity.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Determinant_identities&oldid=791879273"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Determinant_identities
This page is based on the copyrighted Wikipedia article "Determinant identities"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA