Lecture 23: Operations using Partition Matrix or Block Matrix | GATE DA Linear Algebra Module-2

Summary

The lecture develops additional operations on partitioned (block) matrices, emphasizing how block structure can simplify dimension checks, transposes, inverses, determinants, and eigenvalue calculations.

1. Block dimensions and compatibility

In a block matrix, neighboring block dimensions must be compatible for the indicated addition or multiplication to be defined. If (A) is (p \times q), then (A^T) is (q \times p). Identity blocks are square, so their sizes are determined by the adjacent blocks.

For example, consider

[ M=\begin{bmatrix}A&B\C&D\end{bmatrix}. ]

If (B) is (2 \times 5) and (C) is (3 \times 4), compatibility requires:

• (A) to be (2 \times 4),
• (D) to be (3 \times 5),
• (M) to be (5 \times 9).

2. Transposing a block matrix

To transpose a block matrix, transpose each block and swap the off-diagonal block positions:

[ \begin{bmatrix}B&C\E&D\end{bmatrix}^{T} = \begin{bmatrix}B^T&E^T\C^T&D^T\end{bmatrix}. ]

A block matrix with matching diagonal blocks and mutually transposed off-diagonal blocks, such as

[ \begin{bmatrix}I&A\A^T&0\end{bmatrix}, ]

is symmetric, provided its block dimensions are compatible.

3. Block triangular and block diagonal matrices

• A block upper-triangular matrix has zero blocks below the block diagonal.
• A block lower-triangular matrix has zero blocks above the block diagonal.
• A block-diagonal matrix has zero blocks everywhere off the block diagonal.

These structures simplify determinant calculations: the determinant of a block triangular or block-diagonal matrix is the product of the determinants of its diagonal blocks.

4. Inverses of structured block matrices

One method for finding a block inverse is to multiply the given matrix by an unknown block matrix and set the result equal to the block identity matrix. This produces equations for the unknown blocks.

Important cases include:

• Block diagonal:

[ \begin{bmatrix}A&0\0&D\end{bmatrix}^{-1} = \begin{bmatrix}A^{-1}&0\0&D^{-1}\end{bmatrix}, ]

provided (A) and (D) are invertible.

• Block upper-triangular:

[ \begin{bmatrix}A&B\0&D\end{bmatrix}^{-1} = \begin{bmatrix} A^{-1}&-A^{-1}BD^{-1}\ 0&D^{-1} \end{bmatrix}, ]

provided the diagonal blocks are invertible.

• A simple identity-block case:

[ \begin{bmatrix}I&B\0&I\end{bmatrix}^{-1} = \begin{bmatrix}I&-B\0&I\end{bmatrix}. ]

The general inverse of a fully populated block matrix can involve lengthy algebra. The instructor recommends focusing on useful structured cases rather than memorizing a complicated general formula.

5. Determinants of special block matrices

For block upper- or lower-triangular matrices, the determinant is the product of the determinants of the diagonal blocks.

The lecture also discusses

[ \begin{bmatrix}A&B\B&A\end{bmatrix}. ]

When (A) and (B) are compatible square matrices, its determinant can be expressed as

[ \det(A-B)\det(A+B). ]

This identity does not require (A) and (B) to commute.

6. Eigenvalues and eigenvectors

Eigenvalues can be found from the characteristic determinant (\det(A-\lambda I)). If (A) is block diagonal or block triangular, this determinant factors into the characteristic determinants of its diagonal blocks. Consequently, the eigenvalues of the full matrix are the combined eigenvalues of those blocks, including multiplicities.

For a block-diagonal matrix, an eigenvector of one diagonal block can be extended to an eigenvector of the full matrix by placing zeros in the coordinates corresponding to the other block.

For a block upper-triangular matrix, padding an eigenvector of the first diagonal block with zeros works. A vector supported only in the second block does not generally work as an eigenvector, because the upper-right block may contribute a nonzero term.

For

[ \begin{bmatrix}A&B\B&A\end{bmatrix}, ]

the eigenvalues are those of (A+B), together with those of (A-B).

For

[ \begin{bmatrix}A&I\I&A\end{bmatrix}, ]

the eigenvalues are those of (A), each shifted by (+1) and (-1). The lecture applies this result to a GATE CSE 2007-style question.

Correction to a claim in the subtitles: A real matrix does not necessarily have only real eigenvalues. The eigenvalues of the displayed block matrix are those of (A+B) and (A-B), but they need not all be real unless additional conditions ensure this.

Overall lesson

Block matrices are most useful when their structure—especially zeros, identity blocks, or triangular or diagonal form—makes an operation simpler. The recommended approach is to check block dimensions first, then use the structure to reduce the problem to operations on smaller matrices.

Speakers and sources featured

• Speaker: An unnamed instructor from GO Classes for GATE DA.
• David C. Lay (named in the subtitles as “David K. Ley”): a linear algebra textbook is cited as a source for exercises.
• Stack Exchange: cited as the source of several block-matrix questions.
• GATE CSE 2007: a related eigenvalue question is discussed; GATE Overflow is mentioned in connection with its reception.
• Partha Sarthi: an audience response is mentioned during one example.