MSA_Week 1: Block Matrices

1.1 Block Matrix

Definition 1.1.1

Let matrix $A = (a_{ij})$ be a $p \times q$ matrix partitioned into four blocks:

$$ A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$

Where:

• $$ A_{11}: k \times l, \quad A_{12}: k \times (q - l) $$
• $$ A_{21}: (p-k) \times l, \quad A_{22}: (p-k) \times (q-l) $$

This is called the block representation of matrix $A$.

Property 1.1.1 (Addition of Block Matrices)

If matrices $A$ and $B$ have the same dimensions and identical block structures, then:

$$ A + B = \begin{pmatrix} A_{11} + B_{11} & A_{12} + B_{12} \\ A_{21} + B_{21} & A_{22} + B_{22} \end{pmatrix} = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} + \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} $$

Property 1.1.2 (Multiplication of Block Matrices)

Let $A$ be a $p \times q$ matrix and $B$ a $q \times r$ matrix, partitioned appropriately. Then:

$$ AB = \begin{pmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{pmatrix} $$

Property 1.1.3 (Schur Complement)

Suppose $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$ is a square matrix, where $A_{11}$ and $A_{22}$ are square blocks. Then:

1. If $A_{11}$ is invertible, define:

$$ A = A_{11} A_{22\cdot 1}, \quad \text{where} \quad A_{22\cdot 1} = A_{22} - A_{21}A_{11}^{-1}A_{12} $$

2. If $A_{22}$ is invertible, define:

$$ A = A_{11\cdot 2} A_{22}, \quad \text{where} \quad A_{11\cdot 2} = A_{11} - A_{12} A_{22}^{-1} A_{21} $$

The matrices $A_{11\cdot 2}$ and $A_{22\cdot 1}$ are called the Schur complements of matrix $A$.

Schur Complement (Detailed Definition)

Consider a block matrix:

$$ M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} $$

where $A$, $B$, $C$, and $D$ are complex matrices with dimensions:

• $$ A: p \times p, \quad B: p \times q $$
• $$ C: q \times p, \quad D: q \times q $$

If $D$ is invertible, the Schur complement of $D$ in $M$ is defined as:

$$ M / D := A - BD^{-1}C $$

Similarly, if $A$ is invertible, the Schur complement of $A$ in $M$ is:

$$ M / A := D - CA^{-1}B $$

(Reference: Schur complement - Wikipedia)

Property 1.1.4 (Inverse of Block Matrix)

Let $A$ be an invertible square matrix represented as:

$$ A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$

with square blocks $A_{11}$ and $A_{22}$. Then:

1. If $A_{11}$ is invertible:

$$ A^{-1}= \begin{pmatrix} A_{11}^{-1}+A_{11}^{-1}A_{12}A_{22\cdot 1}^{-1}A_{21}A_{11}^{-1} & -A_{11}^{-1}A_{12}A_{22\cdot 1}^{-1} \\ -A_{22\cdot 1}^{-1}A_{21}A_{11}^{-1} & A_{22\cdot 1}^{-1} \end{pmatrix} $$

2. If $A_{22}$ is invertible:

$$ A^{-1}= \begin{pmatrix} A_{11\cdot 2}^{-1} & -A_{11\cdot 2}^{-1}A_{12}A_{22}^{-1} \\ -A_{22}^{-1}A_{21}A_{11\cdot 2}^{-1} & A_{22}^{-1}+A_{22}^{-1}A_{21}A_{11\cdot 2}^{-1}A_{12}A_{22}^{-1} \end{pmatrix} $$