8 Characteristic Roots and Vectors

Some vectors are merely stretched by a matrix rather than turned. This section finds them, and shows what can be done once they are known.

A scalar \(\lambda \) and a non-zero vector \(x\) with \(Ax=\lambda x\) are an eigenvalue and an eigenvector. The search for them reduces to a finite computation: \(\lambda \) is an eigenvalue exactly when \(\det (A-\lambda I)=0\), so the eigenvalues are the roots of a polynomial of degree \(n\).

When there are enough eigenvectors to form a basis, the matrix is diagonalisable — similar to a diagonal matrix — and in that basis it does nothing but scale each coordinate independently. Powers, inverses and polynomials of the matrix then become the same operations on \(n\) numbers.

The section closes with two polynomials attached to a matrix. The Cayley–Hamilton theorem says a matrix satisfies its own characteristic polynomial, and the minimum polynomial refines it: a matrix is diagonalisable precisely when its minimum polynomial has no repeated factor.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.