4.5 Adjoint operators

对照

When $V$ is a vector space with inner product, the adjoint of $L$ can be defined by:

Definition · Adjoint operator

Definition (Adjoint operator). The adjoint of a linear operator $L:V\to V$ is the operator $L^{†}$ satisfying

$$⟨Lv,w⟩=⟨v,L^{†}w⟩$$

for all $v,w\in V$.

Note that mathematicians tend to favor $L^{\ast }$ as notation for the adjoint of $L,$ as opposed to the physicist’s notation $L^{†}$ that we are using.

In terms of explicit matrices, since $l_{Lv}$ is the conjugate-transpose of $Lv$, the matrix for $L^{†}$ will be given by the conjugate-transpose $\overline{L^T}$ of the matrix for $L$:

$$L_{jk}^{†}=\overline{L_{kj}}$$

In the real case, the matrix for the adjoint is just the transpose matrix. We will say that a linear transformation is self-adjoint if $L^{†}=L$ , skew-adjoint if $L^{†}=-L$.

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上一节：4.4 Inner products · 下一节：4.6 Orthogonal and unitary transformations

附录：约定与记号

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正文：英文 · 原著转录

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原书 PDF · 印刷页 41

来源版本：2025-10-20

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