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Determinant identities

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

Contents

Identities

det ( I n ) = 1 where In is the n × n identity matrix.

det ( A T ) = det ( A ) .

det ( A − 1 ) = 1 det ( A ) = det ( A ) − 1 .

For square matrices A and B of equal size,

det ( c A ) = c n det ( A ) 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: det ( A ) = a 1 , 1 a 2 , 2 ⋯ a n , n = ∏ i = 1 n a i , i .

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

L = [ I p 0 − D − 1 C I q ] .

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

M L = [ A B C D ] [ I p 0 − D − 1 C I q ] = [ A − B D − 1 C B 0 D ] = [ I p B D − 1 0 I q ] [ A − B D − 1 C 0 0 D ] .

That is, we have shown that

[ A B C D ] = [ I p B D − 1 0 I q ] [ A − B D − 1 C 0 0 D ] [ I p 0 D − 1 C I q ] ,

References

Determinant identities Wikipedia
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