Symmetric matrices対称行列たいしょうぎょうれつ and orthogonal diagonalization

1Introduction

The important point of this lecture is that a symmetric matrix対称行列たいしょうぎょうれつ is special not only because of its eigenvalues固有値こゆうち, but also because it guarantees orthogonality of eigenvectors in the right sense.

A general matrix may fail to be diagonalizable. Even if it is diagonalizable, its eigenvector basis need not be orthogonal. By contrast, every real symmetric matrix is diagonalizable by an orthonormal basis正規直交基底せいきちょっこうきてい. This is the finite-dimensional spectral theorem.

2Terms and definitions

A real square matrix $A$A is symmetric対称たいしょう if

A complex square matrix $A$A is Hermitianエルミート if

A unitary matrixユニタリ行列ぎょうれつ satisfies

It preserves the complex inner product, lengths, and orthogonality.

A normal matrix正規行列せいきぎょうれつ satisfies

Hermitian and unitary matrices are basic examples of normal matrices.

Orthogonal diagonalization直交対角化ちょっこうたいかくか of a real matrix means writing

with $Q$Q orthogonal and $D$D diagonal. In the complex case one writes

with $U$U unitary.

3Plan

First, symmetry is interpreted through the inner-product identity

Then this identity is used to prove that eigenvectors belonging to distinct eigenvalues are orthogonal.

4Intuitive explanation

A symmetric matrix is a transformation whose input and output contributions are balanced in the inner product. Such a transformation can be decomposed into stretches along mutually orthogonal principal directions. The principal directions are eigenvectors, and the stretch factors are eigenvalues.

5Precise explanation

5.11. Eigenvectors for distinct eigenvalues are orthogonal

Let $A$A be a real symmetric matrix, and suppose

Because $A$A is symmetric,

Substituting the eigenvalue equations gives

Hence

Since $\lambda \ne \mu$\lambda\ne\mu, $⟨u,v⟩=0$\langle u,v\rangle=0, so $u$u and $v$v are orthogonal.

The same argument works for complex Hermitian matrices. If $Au=\lambda u$Au=\lambda u, then

Since $u\ne 0$u\ne0 implies $⟨u,u⟩>0$\langle u,u\rangle>0, we get $\lambda =\stackrel{\_}{\lambda }$\lambda=\overline\lambda, so Hermitian eigenvalues are real.

5.22. Spectral theorem for real symmetric matrices

For a real symmetric matrix $A$A in finite dimensions,

Here $Q$Q is an orthogonal matrix直交行列ちょっこうぎょうれつ, $D$D is diagonal with real eigenvalues, and the columns of $Q$Q are orthonormal eigenvectors of $A$A.

In the complex Hermitian case,

where $U$U is unitary and the diagonal entries of $D$D are real.

5.33. Relation with normal matrices

Over finite-dimensional complex inner product spaces,

Hermitian matrices are the normal matrices whose eigenvalues are real. Real symmetric matrices are the real version of this self-adjoint situation.

6Concrete example

Let

This matrix is symmetric. Take

The corresponding eigenvalues are $3$3 and $1$1. Therefore

and

7Another viewpoint

Ordinary diagonalization chooses a basis that simplifies a matrix. Orthogonal diagonalization makes the stronger claim that this basis can be orthonormal. Because orthogonal and unitary matrices preserve lengths and angles, the change of coordinates does not destroy the inner-product structure.

8Criteria

• A real symmetric matrix is orthogonally diagonalizable.
• A Hermitian matrix is unitarily diagonalizable.
• Over complex finite-dimensional spaces, a matrix is unitarily diagonalizable exactly when it is normal.
• Eigenvectors for distinct eigenvalues of a symmetric or Hermitian matrix are orthogonal.
• Eigenvalues describe stretch factors along principal directions.

9Scope and limitations

Orthogonal diagonalization does not hold for every diagonalizable matrix. It requires symmetry compatible with the inner product. In complex linear algebra, the correct analogue of a real symmetric matrix is a Hermitian matrix, not merely a matrix satisfying $A^T=A$A^T=A.

10Final forms

11In one sentence

A symmetric matrix対称行列たいしょうぎょうれつ decomposes space into mutually orthogonal eigendirections, each with its own stretch factor.

A real symmetric matrix対称行列たいしょうぎょうれつ has enough mutually orthogonal eigenvectors to form an orthonormal basis. Thus it can be written as $A=QD{Q}^T$A=QDQ^T, or equivalently $Q^{T}AQ=D$Q^TAQ=D, where $Q$Q is orthogonal and $D$D is diagonal.

The repeated-eigenvalue case is the main place where mistakes occur. Eigenvectors for different eigenvalues are automatically orthogonal, but inside one eigenspace one may still need Gram-Schmidt to choose an orthonormal basis. Orthogonal diagonalization preserves lengths and angles, which is why it is the right form for quadratic forms and principal-axis calculations.

12Theorem: real and complex spectral theorems

In a finite-dimensional real inner product space, every real symmetric matrix $A=A^T$A=A^T has an orthonormal basis $q_1,\dots ,q_n$q_1,\ldots,q_n of eigenvectors:

Equivalently,

In a finite-dimensional complex inner product space, every normal matrix正規行列せいきぎょうれつ satisfying $A^{*}A=A{A}^{*}$A^*A=AA^* is unitarily diagonalizable:

Hermitian matrices are the special case in which the diagonal entries of $D$D are real.

13Proof idea: from one eigenvector to its orthogonal complement

For a real symmetric matrix, one first chooses a unit eigenvector $q_1$q_1. Then consider the orthogonal complement $q_{1^{\perp }}$q_1^\perp. If $x\in q_{1^{\perp }}$x\in q_1^\perp, then

Thus $Ax\in q_{1^{\perp }}$Ax\in q_1^\perp, so $q_{1^{\perp }}$q_1^\perp is invariant under $A$A. Applying induction to this invariant subspace gives the remaining orthonormal eigenvectors.

The crucial point is that symmetry preserves orthogonal complements. This is exactly what fails for a general matrix.

14Non-example: diagonalizable does not imply orthogonally diagonalizable

The matrix

has distinct eigenvalues $1$1 and $2$2, so it is diagonalizable. But $A^T\ne A$A^T\ne A, and it cannot be orthogonally diagonalized. If $Q^{T}AQ=D$Q^TAQ=D with $Q$Q orthogonal and $D$D diagonal, then $A=QD{Q}^T$A=QDQ^T, which is symmetric. That contradicts $A^T\ne A$A^T\ne A.

This example separates diagonalizability from orthogonal diagonalizability.

15Exercise links

16Related links