# Linear Algebra

ChennaiParis

Vector spaces are defined over a field , while matrix entries can be chosen from elements of commutative ring .

## Cayley-Hamilton theorem

are the eigenvalues, which can be determined by solving .

The Cayley-Hamilton theorem states that ; hence the matrix can be factorized as and

Eigenvector 𝕧 can be obtained by solving 𝕧𝕧.

## Determinant

The determinant function assigns to each matrix, a scalar. It is n-linear, alternating, and .

n-linear means: that for each , is linear over the row, keeping all other rows fixed

Alternating means:

1. If two rows in are equal,
2. If is the matrix we get by interchanging two rows of , then

In category-theoretic terms, the determinant is a natural transformation .