Ch 2.4 - Partitioned Matrices

PARTITIONED MATRICES

Key Features

  • Matrix as Column Vectors:

    • A significant aspect of matrix analysis is viewing matrix A as a collection of column vectors rather than merely as a rectangular arrangement of numbers.

    • This perspective facilitates various matrix operations and manipulations.

  • Other Partitions of A:

    • There is a desire to explore different ways to partition matrix A, indicated by horizontal and vertical dividing lines.

    • This exploration is exemplified in subsequent slides.

Example 1

  • Matrix Representation:

    • Matrix A can be represented as a 2×3 partitioned (or block) matrix.

    • The entries of this matrix can be categorized as blocks or submatrices that correspond to different sections of the overall matrix.

ADDITION AND SCALAR MULTIPLICATION

Addition of Partitioned Matrices

  • Conditions for Addition:

    • If matrices A and B have identical dimensions and are partitioned similarly, it's reasonable to define the addition of the corresponding blocks of A and B.

    • The resulting partitioned matrix from the sum A + B maintains the same partition structure, with each block being the sum of the corresponding blocks of A and B.

Scalar Multiplication of Matrices

  • Scalar Multiplication:

    • When multiplying a partitioned matrix by a scalar, the movement is also executed block by block, thereby ensuring the integrity of the partition structure in the computation.

MULTIPLICATION OF PARTITIONED MATRICES

General Rule for Multiplication

  • Row-Column Rule:

    • Partitioned matrices can be multiplied following the traditional row-column rule if the individual block entries are treated as scalars.

    • For a valid product AB, the column partitions of matrix A must align with the row partitions of matrix B.

Example 3

  • Partitioning Overview:

    • In this illustrative example, matrix A has 5 columns that are separated into a partition of 3 columns followed by 2 columns.

    • Correspondingly, matrix B comprises 5 rows which mirror the partitioning of A into a set of 3 rows and a set of 2 rows.

INVERSES OF PARTITIONED MATRICES

Block Upper Triangular Matrices

  • Description and Conditions:

    • A matrix structured in the form ( A = \begin{bmatrix} A{11} & * \ 0 & A{22} \end{bmatrix} ) is defined as block upper triangular.

    • Where ( A{11} ) is of size ( m \times n ) and ( A{22} ) is of size ( p \times q ), with both being invertible matrices, the goal is to derive ( A^{-1} ) based on the invertibility of these blocks.

Example 5

  • Finding the Inverse:

    • When tasked with calculating the inverse for a block upper triangular matrix, it is essential to have a formula connecting the inverses of the blocks individually to the overall matrix inverse.

Block Diagonal Matrices

  • Definition and Invertibility Criteria:

    • A block diagonal matrix is characterized by having zero blocks positioned off the main diagonal of blocks.

    • Such a matrix is deemed invertible if and only if each block located on the diagonal itself is invertible, reinforcing the requirement for the invertibility of the component matrices to conclude the overall matrix's invertibility.