Let \(A\in\mathbb{F}^{\,n\times n}\) be a square matrix and \(\underline{v} \in \mathbb{F}^n\) be a non-zero vector. \(\underline{v}\) is an eigenvector of \(A\) if and only if

\[A\underline{v}=\lambda\underline{v}\]

where \(\lambda\in\mathbb{F}\) is called an eigenvalue of \(A\) if \(\underline{v}\) exists.

The linear transformation \(A\underline{v}\) where \(\underline{v}\) is an eigenvector of \(A\) is equivalent to the scalar transformation \(\lambda\underline{v}.\)

Eigenvalues satisfy the following properties:

\[\sum_{i=1}^n \lambda_i = \mathrm{tr}(A)\] \[\prod_{i=1}^n \lambda_i = \lvert A\rvert\]

The expression \(\lvert A-\lambda I_n\rvert\) is an \(n^{\mathrm{th}}\)-order polynomial in \(\lambda\) called the characteristic polynomial of \(A.\) The roots \(\lvert A-\lambda I_n\rvert=0\) are the eigenvalues of \(A.\)

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

\[\sigma_A=\{\lambda : \lvert A-\lambda I_n\rvert=0\}\]

The characteristic polynomial of \(A\) will admit \(m\) distinct complex roots where \(1\leq m\leq n.\) Therefore the cardinality of the eigen-spectrum \(\sigma_A\) is \(m.\)

For each eigenvalue \(\lambda_i\in\sigma_A,\) the non-zero solution to

\[\left(A-\lambda_i I_n\right)\underline{v_i}=\underline{0}\]

yields the corresponding eigenvector \(\underline{v_i}\) of \(A.\)

The eigen-space \(\mathcal{E}_{\lambda}\) of \(A\) with respect to eigenvalue \(\lambda\) is the vector space that is spanned by all eigenvectors of \(A.\)

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

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

The algebraic and geometric multiplicities are related by:

\[1\leq\gamma_A\left(\lambda\right)\leq\mu_A\left(\lambda\right)\leq n\,\,\forall\,\lambda\in\sigma_A\]

which means that while distinct eigenvalues correspond to linearly independent eigenvectors, repeated eigenvalues do no necessarily correspond to multiple linearly independent eigenvectors.

\(A\) will admit \(n\) linearly independent eigenvectors if and only if:

\[\gamma_A\left(\lambda_i\right)=\mu_A\left(\lambda_i\right)\,\,\forall\,\lambda_i\in\sigma_A.\]

\(A\) will admit fewer than \(n\) linearly independent eigenvectors if and only if:

\[\gamma_A\left(\lambda_i\right)<\mu_A\left(\lambda_i\right)\,\,\exists\,\lambda_i\in\sigma_A\]

and \(A\) is referred to as defective.

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

Let \(A\) be a non-defective matrix with \(n\) linearly independent eigenvectors \(\underline{v}_1,\ldots,\underline{v}_n\) and corresponding eigenvalues \(\lambda_1,\ldots,\lambda_n.\) The following matrices can be constructed:

\[P=\begin{bmatrix} \uparrow & & \uparrow \\ \underline{v}_1 & \ldots & \underline{v}_n \\ \downarrow & & \downarrow \end{bmatrix}\] \[\Lambda=\begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \lambda_n \end{bmatrix}\]

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

The diagonalisation of \(A\) is the decomposition:

\[A=P\Lambda P^{-1}\]