Learn Spectra and Spectral Decomposition | Spectral Foundations in Linear Algebra

The spectrum of a matrix is the set of its eigenvalues. If you have a square matrix $A$ , and you have already learned to find its eigenvalues and eigenvectors, you can now refer to the collection of all eigenvalues as the spectrum of $A$ . This spectrum provides deep insight into the structure and behavior of the linear transformation represented by the matrix.

Definition

Spectral Theorem for Symmetric Matrices:
If $A$ is a real symmetric matrix, then all its eigenvalues are real, and there exists an orthogonal matrix $Q$ such that $A = Q\Lambda Q^\top$ , where $\Lambda$ is a diagonal matrix whose entries are the eigenvalues of $A$ . This means the spectrum of $A$ consists entirely of real numbers, and $A$ can be represented as a sum of projections onto its eigenvectors.

Intuition: Diagonalization as Change of Basis

When you diagonalize a matrix, you are essentially finding a new basis in which the linear transformation acts by simply stretching or shrinking along each basis direction. In this new basis, the matrix becomes diagonal, and its action is easy to interpret: each coordinate is scaled by the corresponding eigenvalue.

Diagonalization means rewriting the matrix so its structure is clear and simple;
In the new basis, the transformation is just scaling, not mixing directions;
Each eigenvector points in a direction that is unchanged except for scaling by its eigenvalue.

Formal Statement: Spectral Decomposition and the Spectral Theorem

The spectral decomposition of a real symmetric matrix $A$ expresses it as $A = Q\Lambda Q^\top$ , where $Q$ is an orthogonal matrix whose columns are the normalized eigenvectors of $A$ , and $\Lambda$ is a diagonal matrix of eigenvalues (the spectrum).

This is a direct consequence of the Spectral Theorem;
Any symmetric matrix can be "untangled" into independent, one-dimensional actions along its eigenvector directions;
The action of $A$ becomes transparent: each direction is scaled by its corresponding eigenvalue.

Spectral decomposition has important consequences for understanding how matrices behave when raised to powers or when functions are applied to them. If a symmetric matrix $A$ is diagonalizable as $A = Q\Lambda Q^\top$ , then computing powers like $A^k$ becomes straightforward: $A^k = Q\Lambda^k Q^\top$ , where $\Lambda^k$ is simply the diagonal matrix with each eigenvalue raised to the k-th power. This makes it much easier to analyze the long-term behavior of repeated transformations. Similarly, for any function $f$ applied to $A$ , such as the exponential or square root, you can compute $f(A) = Q f(\Lambda) Q^\top$ , where $f(\Lambda)$ is the diagonal matrix with $f$ applied to each eigenvalue. This diagonalization approach reveals why the spectrum is so powerful in simplifying complex matrix operations.

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Section 1. Chapter 2

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