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Chapter 7: Problem 4

Show that if \(A\) and \(B\) are block diagonal matrices with the same blocks, but in possibly different order, then \(A\) and \(B\) are similar.

Short Answer

Expert verified
Given block diagonal matrices A and B with the same blocks M1, M2, ..., Mk in possibly different orders, we can define a permutation matrix P such that B = PAP^(-1). By constructing P using the permutation that orders the blocks in matrix B and computing PAP^(-1), we can show that A and B are similar matrices.

Step by step solution

01

Write the given block diagonal matrices A and B

Let's denote the blocks in matrices A and B as follows: \[ A = \begin{pmatrix} M1 & 0 & \cdots & 0 \\ 0 & M2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & Mk \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} M_{\pi(1)} & 0 & \cdots & 0 \\ 0 & M_{\pi(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & M_{\pi(k)} \end{pmatrix} \] where M1, M2, ..., Mk are the common blocks between both matrices and \(\pi\) is a permutation that denotes the order of the blocks in matrix B.
02

Define the permutation matrix P

Define the permutation matrix P as follows: \[ P = \begin{pmatrix} I_{\pi(1)} & 0 & \cdots & 0 \\ 0 & I_{\pi(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & I_{\pi(k)} \end{pmatrix} \] where I_n denotes the nxn identity matrix.
03

Compute PAP^(-1)

Let's compute the product PAP^(-1), keeping in mind the block structure of the matrices. For block diagonal matrices, the inverse of the matrix is the block diagonal matrix with the inverses of the individual blocks, so P^(-1) can be written as: \[ P^(-1) = \begin{pmatrix} I_{\pi^{-1}(1)} & 0 & \cdots & 0 \\ 0 & I_{\pi^{-1}(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & I_{\pi^{-1}(k)} \end{pmatrix} \] Now compute PAP^(-1): \[ PAP^(-1) = \begin{pmatrix} I_{\pi(1)} & 0 & \cdots & 0 \\ 0 & I_{\pi(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & I_{\pi(k)} \end{pmatrix} \begin{pmatrix} M1 & 0 & \cdots & 0 \\ 0 & M2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & Mk \end{pmatrix} \begin{pmatrix} I_{\pi^{-1}(1)} & 0 & \cdots & 0 \\ 0 & I_{\pi^{-1}(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & I_{\pi^{-1}(k)} \end{pmatrix} \]
04

Simplify PAP^(-1)

Simplify the product using the properties of the identity matrix and the block structure of the matrices: \[ PAP^(-1) = \begin{pmatrix} M_{\pi(1)} & 0 & \cdots & 0 \\ 0 & M_{\pi(2)} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & M_{\pi(k)} \end{pmatrix} \] which matches the matrix B. So, we have shown that B = PAP^(-1) and the matrix P is indeed invertible, which means A and B are similar matrices.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Block Diagonal Matrices
When exploring the concept of block diagonal matrices, think of a matrix as being partitioned into smaller 'blocks' of matrices arranged diagonally with zeroes filling in the non-block areas. Such matrices take the form of
\[ \begin{pmatrix} M1 & 0 & \cdots & 0 \ 0 & M2 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & Mk \end{pmatrix} \]
where each \( M_i \) denotes a square matrix and the zeroes represent blocks of zero entries. The beauty of block diagonal matrices lies in their simplicity for computation, particularly when performing matrix operations like inversion or finding eigenvalues. Each block can be treated independently, which simplifies these complex operations significantly.
Permutation Matrix
Permutation matrices serve as the perfect tool to rearrange the blocks within a block diagonal matrix without altering its inherent properties.
A permutation matrix is a square matrix where in each row and column, one and only one element is '1' and all other elements are '0'. Its primary role is to permute (reorder) the rows or columns of another matrix. For instance,
\[ P = \begin{pmatrix} I_{\pi(1)} & 0 & \cdots & 0 \ 0 & I_{\pi(2)} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & I_{\pi(k)} \end{pmatrix} \]
Here, \( I_{n} \) indicates an identity matrix of order \( n \), and \( \pi \) represents the permutation function specifying the reordering. The charm of permutation matrices is their ability to retain the original matrix's quasi-diagonal form while reordering, which can be seen in applications for matrix similarity transformations.
Matrix Similarity
Matrix similarity is a relation that reflects a profound structural connection between matrices. We say matrices \( A \) and \( B \) are similar if there exists an invertible matrix \( P \) such that
\[ B = PAP^{-1} \]
This transformation keeps the eigenvalues of the matrix intact while the eigenvectors are transformed by P. In the context of block diagonal matrices, similarity implies the same block structures can be rearranged without impacting their algebraic properties. For a pair of block diagonal matrices with the same blocks in different orders, we can construct a permutation matrix \( P \) that, when applied as shown in the exercise solution, demonstrates that these matrices share a similarity relationship. This concept is crucial in understanding the fundamentals of linear transformations and can be applied in various fields, such as quantum mechanics and vibration analysis.

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Most popular questions from this chapter

Prove that the minimal polynomial of \(\tau \in \mathcal{L}(V)\) is the least common multiple of its elementary divisors.

We have seen that any \(\tau \in \mathcal{L}(V)\) can be used to make \(V\) into an \(F[x]\) module. Does every module \(V\) over \(F[x]\) come from some \(\tau \in \mathcal{L}(V)\) ? Explain.

Let \(\tau \in \mathcal{L}(V)\) have minimal polynomial $$ m_{r}(x)=p_{1}^{\epsilon_{1}}(x) \cdots p_{n}^{\epsilon_{1}}(x) $$ The Structure of a Linear Operator 183 where \(p_{i}(x)\) are distinct monic primes. Prove that the following are equivalent: a) \(V_{\tau}\) is \(\tau\)-cyclic. b) \(\operatorname{deg}\left(m_{\tau}(x)\right)=\operatorname{dim}(V)\). c) The elementary divisors of \(\tau\) are the prime power factors \(p_{i}^{c_{i}}(x)\) and so $$ V_{\tau}=\left\langle\left\langle v_{1}\right\rangle\right\rangle \oplus \cdots \oplus\left\langle\left\langle v_{k}\right\rangle\right\rangle $$ is a direct sum of \(\tau\)-cyclic submodules \(\left\langle\left\langle v_{i}\right\rangle\right\rangle\) of order \(p_{i}^{c_{i}}(x)\).

Let \(\tau \in \mathcal{L}(V)\) where \(V\) is finite-dimensional. Describe conditions on the minimal polynomial of \(\tau\) that are equivalent to the fact that the elementary divisor version of the rational canonical form of \(\tau\) is diagonal. What can you say about the elementary divisors?

Prove that a matrix \(A \in \mathcal{M}_{n}(F)\) is nonderogatory if and only if it is similar to a companion matrix.
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