Let A∈F n×nA\in\mathbb{F}^{\,n\times n} be a square matrix and v‾∈Fn\underline{v} \in \mathbb{F}^n be a non-zero vector. v‾\underline{v} is an eigenvector of AA if and only if

Av‾=λv‾A\underline{v}=\lambda\underline{v}

where λ∈F\lambda\in\mathbb{F} is called an eigenvalue of AA if v‾\underline{v} exists.

The linear transformation Av‾A\underline{v} where v‾\underline{v} is an eigenvector of AA is equivalent to the scalar transformation λv‾.\lambda\underline{v}.

Let λ1,…,λn∈C\lambda_1,\ldots,\lambda_n \in \mathbb{C} be the eigenvalues of A,A, repeated according to their algebraic multiplicity (defined below). The eigenvalues satisfy the following properties:

∑i=1nλi=tr(A)\sum_{i=1}^n \lambda_i = \mathrm{tr}(A) ∏i=1nλi=∣A∣\prod_{i=1}^n \lambda_i = \lvert A\rvert

The expression ∣A−λIn∣\lvert A-\lambda I_n\rvert is a polynomial of degree nn in λ\lambda called the characteristic polynomial of A.A. The roots of ∣A−λIn∣=0\lvert A-\lambda I_n\rvert=0 are the eigenvalues of A.A.

The set of all distinct eigenvalues of AA is called the eigen-spectrum σA\sigma_A of A.A.

σA={λ:∣A−λIn∣=0}\sigma_A=\{\lambda : \lvert A-\lambda I_n\rvert=0\}

The characteristic polynomial of AA will admit mm distinct complex roots where 1≤m≤n.1\leq m\leq n. Therefore, taking eigenvalues over C,\mathbb{C}, the cardinality of the eigen-spectrum σA\sigma_A is m.m. A real matrix can have complex eigenvalues: the rotation matrix [0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} has eigenvalues ±i\pm i and no real eigenvalues. The results below that count nn eigenvalues assume that all roots of the characteristic polynomial lie in F,\mathbb{F}, which always holds for F=C.\mathbb{F}=\mathbb{C}.

For each eigenvalue λi∈σA,\lambda_i\in\sigma_A, any non-zero solution to

(A−λiIn)vi‾=0‾\left(A-\lambda_i I_n\right)\underline{v_i}=\underline{0}

is an eigenvector vi‾\underline{v_i} of AA corresponding to λi.\lambda_i.

The eigen-space Eλ\mathcal{E}_{\lambda} of AA with respect to eigenvalue λ\lambda is the vector space that is spanned by all eigenvectors of AA corresponding to λ.\lambda. Equivalently, it is the kernel of A−λIn,A-\lambda I_n, which contains these eigenvectors together with the zero vector:

Eλ={v‾∈Fn:(A−λIn)v‾=0‾}\mathcal{E}_{\lambda}=\{\underline{v} \in \mathbb{F}^n : \left(A-\lambda I_n\right)\underline{v}=\underline{0}\}

The algebraic multiplicity μA(λ)\mu_A\left(\lambda\right) of λ\lambda with respect to the matrix AA is the number of times λ\lambda appears as a root of the characteristic polynomial of A.A. The sum of the algebraic multiplicities of all eigenvalues of AA is n.n.

The geometric multiplicity γA(λ)\gamma_A\left(\lambda\right) of λ\lambda with respect to the matrix AA is the maximum number of linearly independent eigenvectors of AA corresponding to λ\lambda and is equivalent to the dimension of the eigen-space of AA with respect to λ.\lambda.

The algebraic and geometric multiplicities are related by:

1≤γA(λ)≤μA(λ)≤n  ∀ λ∈σA1\leq\gamma_A\left(\lambda\right)\leq\mu_A\left(\lambda\right)\leq n\,\,\forall\,\lambda\in\sigma_A

Eigenvectors corresponding to distinct eigenvalues are always linearly independent, but this inequality shows that a repeated eigenvalue does not necessarily correspond to multiple linearly independent eigenvectors.

AA will admit nn linearly independent eigenvectors if and only if:

γA(λi)=μA(λi)  ∀ λi∈σA.\gamma_A\left(\lambda_i\right)=\mu_A\left(\lambda_i\right)\,\,\forall\,\lambda_i\in\sigma_A.

AA will admit fewer than nn linearly independent eigenvectors if and only if:

γA(λi)<μA(λi)  for some  λi∈σA\gamma_A\left(\lambda_i\right)<\mu_A\left(\lambda_i\right)\,\,\mathrm{for}\,\mathrm{some}\,\,\lambda_i\in\sigma_A

and AA is referred to as defective.

For a non-defective matrix A,A, the nn linearly independent eigenvectors will form a basis for Fn\mathbb{F}^n and is called an eigen-basis.

Let AA be a non-defective matrix with nn linearly independent eigenvectors v‾1,…,v‾n\underline{v}_1,\ldots,\underline{v}_n and corresponding eigenvalues λ1,…,λn.\lambda_1,\ldots,\lambda_n. The following matrices can be constructed:

P=[↑↑v‾1…v‾n↓↓]P=\begin{bmatrix} \uparrow & & \uparrow \\ \underline{v}_1 & \ldots & \underline{v}_n \\ \downarrow & & \downarrow \end{bmatrix} Λ=[λ1000⋱000λn]\Lambda=\begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n \end{bmatrix}

where PP is the eigenvectors in columnar form stacked horizontally to form a square matrix and Λ\Lambda is a diagonal matrix with eigenvalues along the diagonal.

The diagonalisation of AA is the decomposition:

A=PΛP−1A=P\Lambda P^{-1}