Determinant identities

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

Identities

$\det(I_n) = 1$ where I_n is the n × n identity matrix.

$\det(A^{\rm T}) = \det(A).$

$\det(A^{-1}) = \frac{1}{\det(A)}=\det(A)^{-1}.$

For square matrices A and B of equal size,

$\det(AB) = \det(A)\det(B).$

$\det(cA) = c^n\det(A)$ for an n × n matrix.

If A is a triangular matrix, i.e. a_i,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries: $\det(A) = a_{1,1} a_{2,2} \cdots a_{n,n} = \prod_{i=1}^n a_{i,i}.$

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

$L=\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right].$

Here I_p 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

$\begin{align} ML &= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right] = \left[\begin{matrix} A-BD^{-1}C & B \\ 0 & D \end{matrix}\right] \\ &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right]. \end{align}$

That is, we have shown that

$\begin{align} \left[\begin{matrix} A & B \\ C & D \end{matrix}\right] &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right] \left[ \begin{matrix} I_p & 0 \\ D^{-1}C & I_q \end{matrix}\right], \end{align}$