# Linear Algebra

Vector spaces are defined over a field \(\mathcal{F}\), while matrix entries can be chosen from elements of commutative ring \(\mathcal{K}\).

## Cayley-Hamilton theorem

\[A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\]

\(\lambda = 1, 3\) are the eigenvalues, which can be determined by solving \(det(A - \lambda I) = 0\).

The Cayley-Hamilton theorem states that \((A - \lambda_1 I) \ldots (A - \lambda_n I) = 0\); hence the matrix can be factorized as \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) and \(\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}\)

Eigenvector \(\mathbb{v}\) can be obtained by solving \(A \mathbb{v} = \lambda \mathbb{v}\).

## Determinant

The determinant function \(D\) assigns to each \(n \times n\) matrix, a scalar. It is n-linear, alternating, and \(D(I) = 1\).

n-linear means: that for each \(i\), \(D\) is linear over the \(i^{th}\) row, keeping all other rows fixed \[D(\alpha_i + c\alpha_{i'}) = D(\alpha_i) + cD(\alpha_{i'})\]

Alternating means:

- If two rows in \(A\) are equal, \(D(A) = 0\)
- If \(A'\) is the matrix we get by interchanging two rows of \(A\), then \(D(A') = -D(A)\)

In [category-theoretic terms](/ct/1#natural-transformation), the determinant is a natural transformation \(\tau : \textbf{CRng} \overset{\bullet}{\rightarrow} \textbf{Grp}\).