Introduction to Jordan normal formジョルダン標準形ひょうじゅんけい

1Introduction

The key point of this lecture is that Jordan normal formジョルダン標準形ひょうじゅんけい does not give up on matrices that cannot be diagonalized. Instead, it organizes them into a form as close to diagonal as possible.

Diagonalization対角化たいかくか is the ideal case in which eigenvectors固有こゆうベクトル alone form a basis基底きてい. When there are not enough eigenvectors固有こゆうベクトル, generalized eigenvectors一般化いっぱんか固有こゆうベクトル complete the missing directions, and the matrix becomes a direct sum of Jordan blocks.

2Terms and definitions

A generalized eigenvector一般化いっぱんか固有こゆうベクトル for an eigenvalue固有値こゆうち $\lambda$\lambda is a nonzero vector $v$v such that

for some positive integer $k$k.

A Jordan blockジョルダンブロック is

It can be written as

where $N$N is nilpotent and has ones just above the diagonal.

3Plan

For each eigenvalue固有値こゆうち $\lambda$\lambda, separate the diagonal part $\lambda I$\lambda I from the nilpotent part $N$N that prevents diagonalization.

4Intuitive explanation

In a diagonal matrix, each basis基底きてい vector is only scaled in its own direction. In a Jordan blockジョルダンブロック, a small coupling to the neighboring direction remains. This coupling is nilpotent: after applying it enough times, it disappears.

Therefore a Jordan blockジョルダンブロック is not fully diagonal, but its structure is still explicit: eigenvalue固有値こゆうち plus nilpotent drift along a chain.

5Precise explanation

5.11. Standard form

Over an algebraically closed field such as $\mathbb{C}$\mathbb C, every square matrix正方行列せいほうぎょうれつ $A$A has a Jordan normal formジョルダン標準形ひょうじゅんけい: there exists an invertible可逆かぎゃく matrix $P$P such that

where $J$J is a block diagonal matrix whose blocks are Jordan blocks.

More explicitly,

The form is unique up to reordering of the blocks.

5.22. Relation with diagonalization

If every Jordan blockジョルダンブロック has size $1$1, then the Jordan form is diagonal. Thus

5.33. Relation with the minimal polynomial最小多項式さいしょうたこうしき

If $s_{\lambda }$s_\lambda is the largest Jordan blockジョルダンブロック size for eigenvalue固有値こゆうち $\lambda$\lambda, then the minimal polynomial最小多項式さいしょうたこうしき contains the factor

Thus the exponent in the minimal polynomial最小多項式さいしょうたこうしき measures the obstruction to diagonalization.

5.44. Jordan chains

A Jordan blockジョルダンブロック corresponds to a chain of generalized eigenvectors一般化いっぱんか固有こゆうベクトル:

Here $v_1$v_1 is an eigenvector固有こゆうベクトル, and $v_2,\dots ,v_k$v_2,\ldots,v_k are generalized eigenvectors一般化いっぱんか固有こゆうベクトル. Using such chains as a basis基底きてい produces Jordan blocks in the matrix representation.

5.55. Generalized eigenspaces and block sizes

The generalized eigenspace一般化固有空間いっぱんかこゆうくうかん for $\lambda$\lambda is

for sufficiently large $N$N; in finite dimension one may take $N=n$N=n. This space collects all Jordan blocks for $\lambda$\lambda.

The dimension of

is the number of Jordan blocks for $\lambda$\lambda. The sequence of dimensions of

records how chains grow and determines the block sizes.

6Concrete examples

For

we have $A=J_2\left(1\right)$A=J_2(1). The only eigenvalue固有値こゆうち is $1$1, and

so there is only one eigenvector固有こゆうベクトル direction. But

With $v_1=e_1$v_1=e_1 and $v_2=e_2$v_2=e_2,

Thus $v_1,v_2$v_1,v_2 form a Jordan chain of length $2$2.

For a $3$3 by $3$3 block,

Here

Therefore

and the minimal polynomial最小多項式さいしょうたこうしき is

7How to read AM, GM, and blocks

For an eigenvalue固有値こゆうち $\lambda$\lambda:

• The algebraic multiplicity代数的重複度だいすうてきちょうふくど is the sum of the sizes of all Jordan blocks for $\lambda$\lambda.
• The geometric multiplicity幾何的重複度きかてきちょうふくど is the number of Jordan blocks for $\lambda$\lambda.
• The exponent of $(t-\lambda )$(t-\lambda) in the minimal polynomial最小多項式さいしょうたこうしき is the largest block size for $\lambda$\lambda.

Thus AM equals GM exactly when all blocks for that eigenvalue固有値こゆうち have size $1$1.

8Numerical warning

Jordan normal formジョルダン標準形ひょうじゅんけい is extremely useful for theoretical classification, but it is numerically unstable. Small perturbations can split or merge apparent Jordan behavior. In practical computation, Schur decomposition or SVD is often preferred.

9Criteria

• When eigenvectors固有こゆうベクトル are insufficient, introduce generalized eigenvectors一般化いっぱんか固有こゆうベクトル.
• A matrix is diagonalizable exactly when all Jordan blocks have size $1$1.
• The largest block size is the exponent in the minimal polynomial最小多項式さいしょうたこうしき.
• Jordan normal formジョルダン標準形ひょうじゅんけい is unique up to block order.
• The result is naturally stated over a field where the characteristic polynomial特性多項式とくせいたこうしき splits.

10Scope and limitations

Jordan normal formジョルダン標準形ひょうじゅんけい is stated over fields where the characteristic polynomial特性多項式とくせいたこうしき splits into linear factors. Over the real numbers, complex eigenvalues固有値こゆうち may be missing, so one often extends to $\mathbb{C}$\mathbb C.

11Precise form of the existence theorem

Over an algebraically closed field, for example $\mathbb{C}$\mathbb C, every square matrix正方行列せいほうぎょうれつ $A$A has a Jordan normal formジョルダン標準形ひょうじゅんけい. That is, there exists an invertible matrix可逆行列かぎゃくぎょうれつ $P$P such that

This form is unique up to the order of the blocks. The theorem says that even a non-diagonalizable matrix still has a standard form. Over the real numbers alone, complex eigenvalues固有値こゆうち may be missing, so the usual setting is $\mathbb{C}$\mathbb C.

12Final forms

13Generalized eigenspaces and block sizes

The generalized eigenspace一般化固有空間いっぱんかこゆうくうかん for an eigenvalue固有値こゆうち $\lambda$\lambda is

where $N$N is sufficiently large; in finite dimension one may take $N=n$N=n. The space $G_{\lambda }$G_\lambda collects all Jordan blocks for $\lambda$\lambda.

The dimension

is the number of Jordan blocks for $\lambda$\lambda. The dimensions of

measure how chains of length up to $r$r accumulate, and this growth recovers the block sizes.

14In one sentence

Jordan normal formジョルダン標準形ひょうじゅんけい decomposes a non-diagonalizable matrix into eigenvalue固有値こゆうち parts and nilpotent parts.

15Calculation example: a $3$3 by $3$3 Jordan block

Let

Then

Thus

so the minimal polynomial最小多項式さいしょうたこうしき is

The only eigenvalue固有値こゆうち is $2$2, but the nilpotent part has length $3$3, so the structure is far from diagonalizable.

16Exercise link

17How to read AM, GM, and blocks

For an eigenvalue固有値こゆうち $\lambda$\lambda:

• The algebraic multiplicity代数的重複度だいすうてきちょうふくど is the sum of the sizes of all Jordan blocks for $\lambda$\lambda.
• The geometric multiplicity幾何的重複度きかてきちょうふくど is the number of Jordan blocks for $\lambda$\lambda.
• The exponent of $(t-\lambda )$(t-\lambda) in the minimal polynomial最小多項式さいしょうたこうしき is the largest block size for $\lambda$\lambda.

Thus AM equals GM exactly when all blocks for that eigenvalue固有値こゆうち have size $1$1, meaning that the corresponding part is diagonalizable.

18Warning: theory versus numerical computation

Jordan normal formジョルダン標準形ひょうじゅんけい is powerful for theoretical classification. However, for matrices with close eigenvalues固有値こゆうち or small perturbations, Jordan block sizes can change unstably. Practical numerical computation often prefers Schur decomposition or SVD.

19Final forms

20In one sentence

• Jordan normal formジョルダン標準形ひょうじゅんけい describes a non-diagonalizable matrix行列ぎょうれつ by decomposing it into eigenvalue固有値こゆうち parts and nilpotent parts.

21Exercise link

22Related links