Eigenvalues & Eigenvectors Notes

Last reviewed 16 Sept 2026 · 7 min read

Definition

For a square matrix $A$ , a non-zero vector $X$ is an eigenvector and the scalar $λ$ the corresponding eigenvalue (characteristic root, latent root) if

A X = λ X

i.e. $A$ only stretches or shrinks $X$ without changing its direction.

Characteristic equation

(A - λ I) X = O has non-trivial solutions iff ∣ A - λ I ∣ = 0

$∣ A - λ I ∣ = 0$ is the characteristic equation, a polynomial of degree $n$ with $n$ eigenvalues (counting multiplicity, possibly complex).
For a $2 \times 2$ matrix: $λ^{2} - (trace) λ + ∣ A ∣ = 0$ .
For a $3 \times 3$ matrix: $λ^{3} - (trace) λ^{2} + (sum of principal minors of order 2) λ - ∣ A ∣ = 0$ .

Eigenvectors for each $λ$ are found by solving $(A - λ I) X = O$ ; any non-zero multiple of an eigenvector is also an eigenvector.

Properties of eigenvalues

Property	Statement
Sum	$\sum λ_{i} =$ trace of $A$ (sum of diagonal elements)
Product	$\prod λ_{i} = ∣ A ∣$ → $A$ is singular iff at least one eigenvalue is zero
Triangular/diagonal matrices	Eigenvalues are the diagonal elements
Transpose	$A$ and $A^{T}$ have the same eigenvalues
Scalar multiple	$k A$ has eigenvalues $k λ$
Powers	$A^{m}$ has eigenvalues $λ^{m}$ (same eigenvectors)
Inverse	$A^{- 1}$ has eigenvalues $1/ λ$ (same eigenvectors)
Polynomial	$p (A)$ has eigenvalues $p (λ)$ , e.g. $A^{2} + 2 A + I$ → $λ^{2} + 2 λ + 1$
Shift	$A - k I$ has eigenvalues $λ - k$
Adjoint	$adj A$ has eigenvalues $∣ A ∣/ λ$
Similar matrices ( $B = P^{- 1} A P$ )	Same eigenvalues (and characteristic polynomial)
Distinct eigenvalues	Corresponding eigenvectors are linearly independent

Special matrices

Matrix	Eigenvalues	Eigenvectors
Real symmetric	All real	Eigenvectors of distinct eigenvalues are orthogonal; always diagonalisable
Skew-symmetric (real)	Zero or purely imaginary	—
Orthogonal	Unit modulus ( $∣ λ ∣ = 1$ ); if $λ$ is an eigenvalue, so is $1/ λ$
Hermitian	Real
Idempotent	0 or 1
Nilpotent	All zero
Involutory	±1

Algebraic and geometric multiplicity

Algebraic multiplicity (AM) — number of times $λ$ repeats as a root.
Geometric multiplicity (GM) — number of linearly independent eigenvectors for $λ$ $= n - ρ (A - λ I)$ .
$1 \leq GM \leq A M$ ; a matrix is diagonalisable iff GM = AM for every eigenvalue.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation.

If $∣ A - λ I ∣ = λ^{n} + c_{1} λ^{n - 1} + \dots + c_{n} = 0$ , then

A^{n} + c_{1} A^{n - 1} + \dots + c_{n} I = O

Uses:

Inverse: multiply by $A^{- 1}$ and rearrange (for non-singular $A$ ).
Higher powers of $A$ expressed in terms of lower powers.

Diagonalisation

If $A$ ( $n \times n$ ) has $n$ linearly independent eigenvectors, form the modal matrix $P$ with eigenvectors as columns:

P^{- 1} A P = D = diag (λ_{1}, λ_{2}, \dots, λ_{n})

Then $A^{m} = P D^{m} P^{- 1}$ — useful for powers.
For a real symmetric matrix, $P$ can be made orthogonal (normalised eigenvectors) → $P^{T} A P = D$ (orthogonal transformation).

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Eigenvalues & Eigenvectors