Characteristic polynomial

Every square matrix is associated with a characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.

Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a₁, a₂, a₃, etc. then the characteristic polynomial will be: (t-a₁)(t-a₂)(t-a₃)…

This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. A scalar λ is an eigenvalue of A if and only if there is an eigenvector v ≠ 0 such that:

or

(where I is the identity matrix). Since v is non-zero, this means that the matrix λ I − A is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det (λ I − A) are the eigenvalues of A, and it is clear that this determinant is a polynomial in λ.