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 \(A.\)

Eigen-space

Algebraic and geometric multiplicities

Eigen-bases

Diagonalisation

Spectral theorem