4.7 Eigenvalues and eigenvectors

对照

We have seen that the matrix for a linear transformation $L$ of a vector space $V$ changes by conjugation when we change our choice of basis of $V$ . To get basis-independent information about $L,$ one considers the eigenvalues of the matrix. Complex matrices behave in a much simpler fashion than real matrices, since in the complex case the eigenvalue equation

$$\det (\lambda 1-L)=0\text{\hspace{1em}(4.5)}$$

can always be factored into linear factors. For an arbitrary $n$ by $n$ complex matrix there will be $n$ solutions (counting repeated eigenvalues with multiplicity). A basis will exist for which the matrix will be in upper triangular form.

The case of self-adjoint matrices $L$ is much more constrained, since transposition relates matrix elements. One has:

Theorem 4.1 (Spectral theorem for self-adjoint matrices)

Given a self-adjoint complex $n$ by $n$ matrix $L$, there exists a unitary matrix $U$ such that

$$UL{U}^{-1}=D$$

where $D$ is a diagonal matrix with entries $D_{jj}={\lambda }_j,{\lambda }_j\in \mathbf{R}$.

Given $L$, its eigenvalues ${\lambda }_j$ are the solutions to the eigenvalue equation 4.5 and $U$ is determined by the eigenvectors. For distinct eigenvalues the corresponding eigenvectors are orthogonal.

This spectral theorem here is a theorem about finite dimensional vector spaces and matrices, but there are analogous theorems for self-adjoint operators on infinite dimensional state spaces. Such a theorem is of crucial importance in quantum mechanics, where for $L$ an observable, the eigenvectors are the states in the state space with well-defined numerical values characterizing the state, and these numerical values are the eigenvalues. The theorem tells us that, given an observable, we can use it to choose distinguished orthonormal bases for the state space by picking a basis of eigenvectors, normalized to length one.

Using the bra-ket notation in this case we can label elements of such a basis by their eigenvalues, so

$$|j⟩=|{\lambda }_j⟩$$

(the ${\lambda }_j$ may include repeated eigenvalues). A general state is written as a linear combination of basis states

$$|\psi ⟩=\sum _j|j⟩⟨j|\psi ⟩$$

which is sometimes written as a “resolution of the identity operator”

$$\sum _j|j⟩⟨j|=1\text{\hspace{1em}(4.6)}$$

Turning from self-adjoint to unitary matrices, unitary matrices can also be diagonalized by conjugation by another unitary. The diagonal entries will all be complex numbers of unit length, so of the form $e^{i{\lambda }_j},{\lambda }_j\in \mathbf{R}$ . For the simplest examples, consider the cases of the groups $SU(2)$ and $U(2)$ . Any matrix in $U(2)$ can be conjugated by a unitary matrix to the diagonal matrix

$$\left(\begin{array}{cc}{e}^{i{\lambda }_1}& 0\\ 0& e^{i{\lambda }_2}\end{array}\right)$$

which is the exponential of a corresponding diagonalized skew-adjoint matrix

$$\left(\begin{array}{cc}i{\lambda }_1& 0\\ 0& i{\lambda }_2\end{array}\right)$$

For matrices in the subgroup SU(2), ${\lambda }_1=-{\lambda }_2=\lambda$ , so in diagonal form an SU(2) matrix will be

$$\left(\begin{array}{cc}{e}^{i\lambda }& 0\\ 0& e^{-i\lambda }\end{array}\right)$$

which is the exponential of a corresponding diagonalized skew-adjoint matrix that has trace zero

$$\left(\begin{array}{cc}i\lambda & 0\\ 0& -i\lambda \end{array}\right)$$

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上一节：4.6 Orthogonal and unitary transformations · 下一节：4.8 For further reading

附录：约定与记号

来源与版本

正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 44、45、46

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837