Determinant identities

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


where In is the n × n identity matrix.

provided exists.

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.

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