Basics of diagonalization対角化たいかくか

1Introduction

The key point of this lecture is that diagonalization対角化たいかくか is not merely a technique for rewriting a matrix行列ぎょうれつ. It is the process of choosing an eigenbasis固有基底こゆうきてい so that the essential action of a linear transformation線型変換せんけいへんかん becomes visible.

A diagonal matrix対角行列たいかくぎょうれつ only multiplies each coordinate direction by a scalar. Therefore, if a complicated matrix becomes diagonal after choosing a better basis基底きてい, the structure of the transformation becomes much easier to understand.

2Terms and definitions

A matrix $A$A is diagonalizable対角化可能たいかくかかのう if there is an invertible matrix $P$P such that

and $D$D is diagonal.

This equation preserves the underlying linear map線型写像せんけいしゃぞう. What changes is the coordinate representation used to read that map. The matrix $A$A is the representation in the standard basis, $D$D is the representation in an eigenbasis, and $P$P converts coordinates in the eigenbasis back to standard coordinates.

An eigenbasis固有基底こゆうきてい is a basis基底きてい consisting entirely of eigenvectors固有こゆうベクトル.

3Plan

First we show that if eigenvectors form a basis, then the matrix representation becomes diagonal. Diagonalization is a change of basis applied to a linear map: the same transformation is rewritten in an eigenbasis. Then we express diagonalizability using geometric and algebraic multiplicities, and finally connect diagonalization to powers, recurrence relations, and differential equations.

4Intuitive explanation

In standard coordinates, a transformation may mix coordinate directions in a complicated way. If we rotate or change the coordinate axes so that they align with the natural directions of stretching, the transformation becomes simple. Aligning the coordinate axes with eigendirections固有方向こゆうほうこう is the idea of diagonalization対角化たいかくか.

5Precise explanation

5.11. Put eigenvectors into a matrix

Suppose $v_1,\dots ,v_n$v_1,\ldots,v_n are linearly independent eigenvectors of $A$A, with corresponding eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$\lambda_1,\ldots,\lambda_n. Define

Then

Indeed, the $j$j-th column of $AP$AP is $A{v}_j={\lambda }_j{v}_j$Av_j=\lambda_jv_j, and the $j$j-th column of $PD$PD is also ${\lambda }_j{v}_j$\lambda_jv_j. Since the $v_j$v_j are linearly independent, $P$P is invertible, so

For example, let

We may take

Then

and $P^{-1}AP=D$P^{-1}AP=D. The linear transformation itself has not changed; only the axes used to describe it have changed. In the eigenbasis, the first axis is scaled by $3$3 and the second by $2$2.

5.22. Conditions for diagonalization対角化たいかくか

For an $n$n-dimensional space, $A$A is diagonalizable exactly when there are $n$n linearly independent eigenvectors.

If all eigenvalues are distinct, their eigenvectors are automatically linearly independent, so $A$A is diagonalizable. This is a sufficient condition, not a necessary one.

More fundamentally,

When the characteristic polynomial splits over the field, $A$A is diagonalizable exactly when every eigenvalue $\lambda$\lambda satisfies

Here $\mathrm{G}\mathrm{M}$\mathrm{GM} is geometric multiplicity and $\mathrm{A}\mathrm{M}$\mathrm{AM} is algebraic multiplicity.

5.33. Why diagonalization is useful

If $A=PD{P}^{-1}$A=PDP^{-1}, then

This makes high powers and repeated applications much easier to compute, because $D^n$D^n is obtained by raising each diagonal entry to the $n$n-th power.

For a system of differential equations

one studies

If $A=PD{P}^{-1}$A=PDP^{-1}, then

and $e^{tD}$e^{tD} is the diagonal matrix whose entries are $e^{t{\lambda }_i}$e^{t\lambda_i}. Thus diagonalization is useful not only for powers of matrices but also for differential equations.

6Criteria

• After finding eigenvalues固有値こゆうち and eigenvectors固有こゆうベクトル, ask whether enough linearly independent eigenvectors exist.
• When large powers of a matrix or recurrence relations appear, consider diagonalization.
• Distinct eigenvalues guarantee diagonalizability. Repeated eigenvalues require checking whether GM equals AM.

7Scope and limitations

Having eigenvalues is not the same as being diagonalizable. For example,

has the eigenvalue $1$1, but only one linearly independent eigenvector. It cannot be diagonalized.

8Final forms

9In one sentence

Diagonalization対角化たいかくか means choosing eigenvectors固有こゆうベクトル as a basis基底きてい so that a linear transformation線型変換せんけいへんかん is rewritten as one scale factor per direction.

10Theorem and proof: diagonalizability and eigenvector bases

For an $n\times n$n\times n matrix $A$A, $A$A is diagonalizable if and only if $F^n$\mathbb F^n has a basis consisting of eigenvectors of $A$A.

First suppose $A=PD{P}^{-1}$A=PDP^{-1}. Let the columns of $P$P be $p_1,\dots ,p_n$p_1,\ldots,p_n, and let the diagonal entries of $D$D be ${\lambda }_1,\dots ,{\lambda }_n$\lambda_1,\ldots,\lambda_n. Since $AP=PD$AP=PD, comparing columns gives

Because $P$P is invertible, its columns form a basis. Hence there is an eigenvector basis.

Conversely, suppose $v_1,\dots ,v_n$v_1,\ldots,v_n is an eigenvector basis and $A{v}_i={\lambda }_i{v}_i$Av_i=\lambda_iv_i. Let

Then $AP=PD$AP=PD column by column. Since $P$P is invertible, $A=PD{P}^{-1}$A=PDP^{-1}.

This theorem shows that diagonalization is exactly the operation of replacing coordinate axes by eigendirections.

11Exercise link

12Related links