DLA Motivation

Section 28.1 Motivation

Subsection 28.1.1 Looking back to look forward

We have discussed how the similarity relation partitions

$\matrixring_n(\C)$ into similarity classes (Subsection 26.3.3). This fact lets us view our study of diagonalization (Chapter 25) in a new context: a matrix is diagonalizable precisely when its similarity class contains a diagonal matrix.

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We focused on diagonal matrices because it is a very simple form. From Subsection 26.5.2, we know that similar matrices share many properties. If a matrix is diagonalizable, we can determine a lot of its properties indirectly by just considering a diagonal matrix to which it is similar.

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What if a matrix not diagonalizable? What is the next simplest form we can look for in the similarity class of that matrix?

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Question 28.1.1.

Is there some sort of special form of matrix that is reasonably simple and of which every similarity class contains at least one example? In other words, can every square matrix be made similar to some particularly simple form of matrix?

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We will pursue two separate threads of the answer to this question in this chapter and the next, and then unify them into a preliminary answer in the chapter after that. And it will take us a few more chapters after that to get to the final, simplest form that answers the question above.

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Remark 28.1.2.

You might now be wondering why we started with diagonal matrices when they are not the simplest form of matrix. Scalar matrices are an even simpler form than diagonal, but each scalar matrix lives in its own similarity class by itself (Proposition 26.5.2), so there was no point in attempting to study “scalarizable” matrices.

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Subsection 28.1.2 Generalizing the diagonalizable case

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Philosophy of inquiry: generalize.

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A frequently-used strategy for incrementally furthering a program of study in mathematics is to try to generalize a previous result: to make the specifications or boundaries of the problem a little less precise or restrictive, and to try to adapt the old solution techniques to the new, less-constrained situation.

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In the spirit of the philosophy above, the next form we study in this chapter is a generalization of diagonal form called block-diagonal form. A matrix of this form can be split into a pattern of submatrices so that each submatrix that is not on the main diagonal of blocks is a zero matrix.

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Here is an example. The matrix

$\begin{equation*} \left[\begin{array}{@{}ccc|r|cc@{}} 1 \amp 2 \amp 3 \amp 0 \amp 0 \amp 0 \\ 4 \amp 5 \amp 6 \amp 0 \amp 0 \amp 0 \\ 7 \amp 8 \amp 9 \amp 0 \amp 0 \amp 0 \\ \hline 0 \amp 0 \amp 0 \amp -1 \amp 0 \amp 0 \\ \hline 0 \amp 0 \amp 0 \amp 0 \amp 1 \amp 2 \\ 0 \amp 0 \amp 0 \amp 0 \amp 3 \amp 4 \\ \end{array}\right] \end{equation*}$

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is in block-diagonal form, where the diagonal blocks, in order, have dimensions

$3 \times 3\text{,}$

$1 \times 1\text{,}$ and

$2 \times 2\text{.}$ (Grid lines have been added to emphasize the block pattern of the matrix.)

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Usually, we do not bother to write all the zeros — you should assume that any blank entries in a matrix are zero. For example, the block-diagonal form of the example matrix above is more clear if we write it as

$\begin{equation*} \begin{bmatrix} 1 \amp 2 \amp 3 \\ 4 \amp 5 \amp 6 \\ 7 \amp 8 \amp 9 \\ \amp \amp \amp -1 \\ \amp \amp \amp \amp 1 \amp 2 \\ \amp \amp \amp \amp 3 \amp 4 \end{bmatrix}\text{.} \end{equation*}$

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How is this form a generalization of diagonal form? Every diagonal matrix could be considered a block-diagonal matrix in which the blocks all happen to be size

$1 \times 1\text{.}$ And as we carry out our study of block-diagonal form, we will find that the eigenvector-eigenvalue method of solution of the diagonalizable case has something in common with the block-diagonal case.