Lagmental Vicfred

The Spectral Theorem Diagonalizes Self-Adjoint Operators by Vicfred

Last updated: Wed 04 December 2024

tags: mathematics, inner products, spectral theorem, quadratic forms

A finite-dimensional self-adjoint operator has an orthonormal eigenbasis and real eigenvalues. The point is to make the formal expression readable enough to audit line by line.

Definitions first

An inner product \(\langle x,y\rangle\) converts algebraic decompositions into orthogonal ones. Self-adjoint maps satisfy \(T=T^\ast\) and have real spectral data.

$$ T=T^\ast $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ T=Q\Lambda Q^\ast,\qquad Q^\ast Q=I,\quad\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)\in\mathbf R^{n\times n} $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ f(T)=Q\operatorname{diag}(f(\lambda_1),\ldots,f(\lambda_n))Q^\ast $$

The reusable statement

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad T=T^\ast,\\[5pt] \mathsf{C}\;&:\quad T=Q\Lambda Q^\ast,\qquad Q^\ast Q=I,\quad\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)\in\mathbf R^{n\times n}. \end{aligned} $$

A nearby false statement

Orthogonal diagonalization requires self-adjointness over the real or complex inner-product setting. A general diagonalizable matrix need not have orthogonal eigenvectors.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] T=Q\Lambda Q^\ast,\qquad Q^\ast Q=I,\quad\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)\in\mathbf R^{n\times n} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sat 14 January 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.