Definition. If M is a square matrix of size n, the characteristic polynomial of M is the polynomial phi_{M} in the variable t that is given by the formula phi_{M}(t) = det ( t . 1_{n} - M ), where 1_{n} denotes the identity matrix.
Properties. Let M be an n \times n matrix.
phi_{M} is a polynomial of degree n.
The coefficient in phi_{M} of degree n is 1.
The constant term in phi_{M}, i.e., the coefficient of degree 0, is (-1)^{n} det (M).
The coefficient in phi_{M} of degree n-1 is the negative of the trace of M. (The trace of a square matrix is the sum of its diagonal elements.)
The roots of the characteristic polynomial of a diagonal matrix are the diagonal entries of the matrix.
Similar matrices have the same characteristic polynomial.
Definition. The characteristic polynomial of a linear map of a finite-dimensional vector space is the characteristic polynomial of its matrix relative to any basis of the vector space. (It is independent of the choice of basis, even though the matrix is not, since the matrices arising from different bases are similar.)
Proposition. The roots of the characteristic polynomial of a linear map are the proper values of the linear map.
Proposition. If the characteristic polynomial of an n \times n matrix has n distinct (real) roots, then the matrix must be similar to a diagonal matrix.
Let A be the 3\times 3 matrix
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| 0 | 1 | 0 |
Find the characteristic polynomial of A.
Find the proper values of the linear transformation f : x -> Ax.
For each proper value c of f find the characteristic subspace of f for c.
Is A similar to a diagonal matrix?
Let f be the linear transformation that is defined by f(x) = M x where M is the 4 \times 4 matrix
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
Find the characteristic polynomial of M.
Find all proper values of f.
For each proper value c of f find the characteristic subspace of f for c.
Is M similar to a diagonal matrix?
Give an example of a 2\times 2 matrix C not similar to a diagonal matrix for which the characteristic polynomial phi_{C} has a real root of multiplicity 2.