Symmetric Matrices and the Spectral Theorem

A real symmetric matrix has real eigenvalues and an orthonormal eigenbasis. That single fact is behind quadratic forms, covariance matrices and principal axes.

ages 20 to 218 practice questionsvideo included

What a learner can do afterwards

Diagonalise a symmetric matrix with an orthogonal matrix
Classify a quadratic form from the signs of its eigenvalues
Explain why symmetry forces eigenvectors for distinct eigenvalues to be orthogonal

1 · Read

A real symmetric matrix has real eigenvalues and an orthonormal eigenbasis. That single fact unlocks the spectral decomposition: A equals Q times D times Q transpose, where Q is orthogonal and D holds the real eigenvalues. Follow the usual pipeline on a symmetric two by two: solve the characteristic equation, then find each eigenspace. For symmetric input the pipeline always succeeds.

Try it together

Diagonalise the symmetric matrix with rows (5, 2) and (2, 8). Its trace is 13 and its determinant is 40 minus 4, which is 36. The characteristic equation is lambda squared minus 13 lambda plus 36 equals 0, with roots 9 and 4. Both are real, as symmetry promises. Normalise the two eigenvectors and stack them as Q, and you hold the spectral decomposition in your hands.

Symmetry also forces eigenvectors for distinct eigenvalues to be orthogonal. Dot the two eigenvectors from the example and you get zero, which is no accident: a short argument shows distinct eigenvalues always give perpendicular directions. That is why Q can be orthogonal at all. The same signs then classify quadratic forms: all positive eigenvalues bowl upward, all negative bowl downward, and mixed signs saddle.

Good to know

This one fact powers covariance matrices and principal axes, so learn to reproduce the orthogonality argument from memory. And always normalise before stacking Q: only then is the transpose the true inverse and the factorisation genuinely spectral.

Symmetry gives real eigenvalues with perpendicular directions, stacked into Q with the values in D.

2 · Watch

Take it off screen

Print a worksheetA4 with an answer key page for grown-ups. No screen, no internet.

Where it sits

Learn first

The Gram-Schmidt Process Eigenvalues and Eigenvectors

This opens up

States as Vectors: Dirac Notation and the State Space Hartree-Fock and What a Basis Set Decides Iterative Methods for Large Linear Systems Classifying Second-Order PDEs and What the Type Decides Principal Components and Dimensionality Reduction Compact Self-Adjoint Operators and the Spectral Theorem Observables as Hermitian Operators and Their Spectra Degenerate Perturbation Theory and the Variational Method First-Order Perturbation Theory The Singular Value Decomposition

Then practise

8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.

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