Diagonalization of Symmetric Matrices

Diagonalization factors a matrix as $A = P D P^{- 1}$ when it has enough independent eigenvectors. For a general matrix that condition can fail, and even when it holds the eigenvector matrix $P$ is usually not convenient: its inverse must be computed, and its columns sit at arbitrary angles. Symmetric matrices avoid both problems at once. They are always diagonalizable, their eigenvectors can always be chosen mutually perpendicular, and the change-of-basis matrix is orthogonal, so its inverse is just its transpose.

For example, $[10 0 - 3]$ and $0 - 1 5 - 1 30 50 - 2$ are symmetric; a matrix with $a_{12} = 4$ but $a_{21} = 6$ is not.

Eigenvectors of a symmetric matrix are orthogonal

Eigenvectors belonging to different eigenvalues are automatically perpendicular. Nothing like this holds for a general matrix.

The proof turns the symmetry $A^{⊤} = A$ into a statement about the dot product. Let $v_{1}, v_{2}$ be eigenvectors for distinct eigenvalues $λ_{1}, λ_{2}$ . Using $u^{⊤} w = u \cdot w$ and moving $A$ across the product with its transpose,

λ_{1} (v_{1} \cdot v_{2}) = (λ_{1} v_{1})^{⊤} v_{2} = (A v_{1})^{⊤} v_{2} = v_{1}^{⊤} A^{⊤} v_{2} = v_{1}^{⊤} A v_{2} = v_{1}^{⊤} (λ_{2} v_{2}) = λ_{2} (v_{1} \cdot v_{2}) .

The middle equality $A^{⊤} = A$ is the only place symmetry enters. Rearranging, $(λ_{1} - λ_{2}) (v_{1} \cdot v_{2}) = 0$ , and since $λ_{1} \neq = λ_{2}$ the eigenvalues cancel, forcing $v_{1} \cdot v_{2} = 0$ . For a general matrix, distinct eigenvalues give only independent eigenspaces; symmetry makes them orthogonal.

A symmetric matrix stretches space along perpendicular axes: the unit circle maps to an ellipse whose axes lie along the orthogonal eigenvectors, scaled by the eigenvalues.

Orthogonal diagonalizability

An orthogonal matrix $P$ has orthonormal columns, which is equivalent to $P^{⊤} P = I$ , so its inverse is $P^{- 1} = P^{⊤}$ .¹ When the eigenvector matrix can be chosen orthogonal, diagonalization needs no inverse.

The condition does more than simplify computation; it characterizes symmetry. One direction is immediate: if $A = P D P^{⊤}$ , then

A^{⊤} = (P D P^{⊤})^{⊤} = P^{⊤⊤} D^{⊤} P^{⊤} = P D P^{⊤} = A,

using $D^{⊤} = D$ for a diagonal matrix. So an orthogonally diagonalizable matrix must be symmetric. The converse is the harder and more useful statement.

For a general matrix, deciding diagonalizability requires inspecting eigenvalue multiplicities case by case. For symmetric matrices the question is settled in advance by a glance at the entries.

The Spectral Theorem

The orthogonal-diagonalizability theorem is one part of a larger statement. The set of eigenvalues of $A$ is its spectrum, and the Spectral Theorem collects four properties of a real symmetric matrix.

Property (a) says the characteristic polynomial of a real symmetric matrix has no complex roots, so the spiral dynamics of complex eigenvalues never occur here. Property (b) rules out the defect that blocks diagonalization in the general case: a symmetric matrix never runs short of eigenvectors. Property (c) is the orthogonal-eigenspaces theorem. Together they guarantee (d): assemble an orthonormal basis inside each eigenspace, and because eigenspaces are orthogonal, the pooled set is an orthonormal eigenvector basis for $R^{n}$ .

Orthogonal diagonalization sends A to a pure scaling D in the eigenvector coordinates: read into the eigenbasis by Pᵀ, scale each axis by its eigenvalue, then read back out by P.

Distinct eigenvalues

When the eigenvalues are distinct, the eigenvectors are automatically orthogonal and only normalization remains.

Repeated eigenvalues

When an eigenvalue is repeated, the eigenvectors returned by row reduction span the right space but need not be perpendicular to one another. Gram–Schmidt applied inside the eigenspace repairs this.

Worked example. Orthogonally diagonalize

A = 3 - 2 4 - 2 62 423, 0 = - λ^{3} + 12 λ^{2} - 21 λ - 98 = - (λ - 7)^{2} (λ + 2) .

The eigenvalue $7$ has multiplicity two. Row reduction of $A - 7 I$ produces two independent eigenvectors,

λ = 7 : v_{1} = 101, v_{2} = - 1/2 10, λ = - 2 : v_{3} = - 1 - 1/2 1 .

Here $v_{1} \cdot v_{2} = - \frac{1}{2} \neq = 0$ , so the two vectors in the $λ = 7$ eigenspace are not orthogonal. Subtracting the projection of $v_{2}$ onto $v_{1}$ ,

z_{2} = v_{2} - \frac{v _{2} \cdot v _{1}}{v _{1} \cdot v _{1}} v_{1} = - 1/2 10 - \frac{- 1/2}{2} 101 = - 1/4 1 1/4 .

Since $z_{2}$ is a linear combination of two eigenvectors for $λ = 7$ , it is itself an eigenvector for $λ = 7$ , and by construction $z_{2} ⊥ v_{1}$ . So ${v_{1}, z_{2}}$ is an orthogonal basis for the two-dimensional eigenspace. The eigenvector $v_{3}$ for the distinct eigenvalue $- 2$ is automatically orthogonal to both by the orthogonal-eigenspaces theorem. Normalizing all three gives an orthonormal eigenvector basis, and $A = P D P^{⊤}$ with $D = diag (7, 7, - 2)$ .

Inside a repeated eigenspace the raw eigenvectors need not be perpendicular; Gram–Schmidt subtracts the projection of v₂ onto v₁ to produce z₂ ⟂ v₁ spanning the same plane.

Spectral decomposition

Writing $A = P D P^{⊤}$ and expanding the product columnwise expresses $A$ as a sum of one-term contributions, one per eigenpair. Let the columns of $P$ be the orthonormal eigenvectors $u_{1}, \dots, u_{n}$ and the diagonal of $D$ the eigenvalues $λ_{1}, \dots, λ_{n}$ . The column–row rule for a product gives

A = P D P^{⊤} = [u_{1} \dots u_{n}] λ_{1} ⋱ λ_{n} u_{1}^{⊤} ⋮ u_{n}^{⊤} = λ_{1} u_{1} u_{1}^{⊤} + λ_{2} u_{2} u_{2}^{⊤} + \dots + λ_{n} u_{n} u_{n}^{⊤} .

Each term $u_{i} u_{i}^{⊤}$ is an $n \times n$ matrix of rank one: every column is a scalar multiple of $u_{i}$ . More is true. The matrix $u_{i} u_{i}^{⊤}$ is the orthogonal projection onto the line spanned by $u_{i}$ , because for any $x$ ,

(u_{i} u_{i}^{⊤}) x = u_{i} (u_{i}^{⊤} x) = (u_{i} \cdot x) u_{i},

which reproduces the projection formula for a unit vector. So the spectral decomposition resolves $A$ into projections onto its eigenvector axes, each scaled by its eigenvalue. Applying $A$ to a vector projects onto each axis, rescales by $λ_{i}$ , and sums.

Spectral decomposition stacks A out of rank-one projection layers: each uᵢuᵢᵀ projects onto one eigenvector axis, scaled by its eigenvalue λᵢ.

The three ways to hold a symmetric matrix each expose a different fact about it.

Form	Expression	What it exposes
Entries	$A$ with $a_{ij} = a_{j i}$	symmetry, at a glance
Orthogonal diagonalization	$A = P D P^{⊤}$	eigenvalues and perpendicular axes
Spectral decomposition	$A = \sum_{i} λ_{i} u_{i} u_{i}^{⊤}$	rank-one projection structure

Orthogonality and computation

The eigenvalues of a symmetric matrix are computed by applying a sequence of similarity transformations built from orthogonal matrices, driving $A$ toward diagonal form while its diagonal entries converge to the eigenvalues. Because orthogonal matrices preserve lengths, rounding errors do not accumulate the way they can under a general similarity, so the process is numerically stable. When $A$ is symmetric the orthogonal factors multiply together into a single orthogonal matrix whose columns are the eigenvectors. A nonsymmetric matrix has no full orthogonal eigenvector set, so after its eigenvalues are found the eigenvectors require separate, less stable work. The numerical eigenvalue methods develop this computation in detail.

Two closure properties fall directly out of the theory. If $A$ is symmetric then $A^{2}$ is symmetric, since $(A^{2})^{⊤} = (A^{⊤})^{2} = A^{2}$ ; and if $A$ is orthogonally diagonalizable then so is $A^{2} = P D^{2} P^{⊤}$ , with the eigenvalues squared and the same eigenvectors. The eigenvector axes are fixed by $A$ ; powers of $A$ only rescale the stretch along them.

Lay, §6.2 — Orthogonal Sets: a square matrix with orthonormal columns satisfies $P^{⊤} P = I$ , hence $P^{- 1} = P^{⊤}$ , and such a matrix preserves lengths and dot products. ↩

Eigenvectors of a symmetric matrix are orthogonal

Orthogonal diagonalizability

The Spectral Theorem

Distinct eigenvalues

Repeated eigenvalues

Spectral decomposition

Orthogonality and computation

Footnotes