Inverse of Block Matrices
The inverse of a partitioned matrix appears so often in research, I decided to collect some of the results into this document. Most of these results come from
- “Inverses of 2x2 Block Matrices” by Tzon-Tzer Lu and Sheng-Hua Shiou
- “The Matrix Cookbook” by Peterson and Pedersen.
\(2 \times 2\) matrices
Given an invertible matrix
\[A = \begin{bmatrix} a & b\\ c & d \end{bmatrix}\]the inverse is
\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b\\ -c & a \end{bmatrix} = \frac{1}{\operatorname{det}(A)} \operatorname{adj}(A)\]General Block Partitioned Matrices
Consider
\[P = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\]where \(A, B, C, D\) are each matrices of compatible sizes.
Then the inverse (if it exists) can be written as
\[P^{-1} = \begin{bmatrix} E & F \\ G & H \end{bmatrix}\]We have different cases based on the invertibility of \(P\). All sizes are assumed to be compatible.
Theorem 1: Assume \(A\) is invertible. The matrix \(P\) is invertible if and only if the schur complement of \(A\),
\[X = D - C A^{-1} B = S(A)\]is invertible, and the inverse is
\[P^{-1} = \begin{bmatrix} A^{-1} + A^{-1} B X^{-1} C A^{-1} & -A^{-1} B X^{-1} \\ -X^{-1} C A^{-1} & X^{-1} \end{bmatrix}\]Proof: See Theorem 2.1 of Lu and Shiou, “Inverses of 2x2 Block Matrices”, 2000
Theorem 2: Assume \(B\) is invertible. The matrix \(P\) is invertible if and only if the schur complement of \(B\),
\[X = C - D B^{-1} A\]is invertible, and the inverse is
\[P^{-1} = \begin{bmatrix} - X^{-1} D B^{-1} & X^{-1} \\ - B^{-1} + B^{-1} A X^{-1} D B^{-1} & - B^{-1} A X^{-1} \end{bmatrix}\]Proof: See Theorem 2.2 of Lu and Shiou, “Inverses of 2x2 Block Matrices”, 2000
Theorem 3: Assume \(C\) is invertible. The matrix \(P\) is invertible if and only if the schur complement of \(C\),
\[X = B - A C^{-1} D\]is invertible, and the inverse is
\[P^{-1} = \begin{bmatrix} -C^{-1} D X^{-1} & C^{-1} + C^{-1} D X^{-1} A C^{-1}\\ X^{-1} & - X^{-1} A C^{-1} \end{bmatrix}\]Proof: See Theorem 2.2 of Lu and Shiou, “Inverses of 2x2 Block Matrices”, 2000
Theorem 4: Assume \(D\) is invertible. The matrix \(P\) is invertible if and only if the schur complement of \(D\),
\[X = A - B D^{-1} C\]is invertible, and the inverse is
\[P^{-1} = \begin{bmatrix} X^{-1} & - X^{-1} B D^{-1}\\ -D^{-1} C X^{-1} & D^{-1} + D^{-1} C X^{-1} B D^{-1} \end{bmatrix}\]Proof: See Theorem 2.1 of Lu and Shiou, “Inverses of 2x2 Block Matrices”, 2000
We can use the above equations to derive some corollaries. These are repeated here, without proof.
Corollary 1: For invertible \(A, D\),
\[P = \begin{bmatrix} A & 0\\ 0 & D \end{bmatrix} \iff P^{-1} = \begin{bmatrix} A^{-1} & 0 \\ 0 & D^{-1} \end{bmatrix}\]Corollary 2: For invertible \(B, C\),
\[P = \begin{bmatrix} 0 & B \\ C & 0 \end{bmatrix} \iff P^{-1} = \begin{bmatrix} 0 & C^{-1}\\ B^{-1} & 0 \end{bmatrix}\]Theorem 2, 3, and corollary 2 are particularly useful in cases like
\[P = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \end{bmatrix}\](which is invertible, \(P^{-1} = P\)).
Corollary 3:
\[P = \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}\]has an inverse if and only if \(A, D\) are invertible. Then,
\[P^{-1} = \begin{bmatrix} A^{-1} & - A^{-1} B D^{-1}\\ 0 & D^{-1} \end{bmatrix}\]Corollary 4:
For \(B\) invertible,
\[P = \begin{bmatrix} A & B \\ 0 & D \end{bmatrix}\]has an inverse if and only if \(X = D B^{-1} A\) is invertible. Then,
\[P^{-1} = \begin{bmatrix} X^{-1} DB^{-1} & - X^{-1}\\ B^{-1} - B^{-1} A X^{-1} D B^{-1} & B^{-1} A X^{-1} \end{bmatrix}\]Very similar expressions also exist for lower triangular matrices.