92. Characteristic Polynomial
For an matrix , the characteristic polynomial is:
where is the identity matrix.
is a polynomial in of degree . Its roots are the eigenvalues of .
92.1. Why the roots are the eigenvalues
By definition, is an eigenvalue of iff there exists a non-zero with , i.e. . A non-trivial solution exists iff is singular iff . So the eigenvalues are exactly the roots of .
Example
Roots: .
The eigenvalues of are and (the diagonal entries — a feature of triangular matrices).
92.2. Structure of the polynomial
For an matrix, with:
- Leading coefficient: (the sign depends on ; some conventions use )
- Coefficient of : — gives the trace
- Constant term: (set )
For matrices:
92.3. Multiplicity
- Algebraic multiplicity of : its multiplicity as a root of .
- Geometric multiplicity of : — the number of linearly independent eigenvectors for .
Always: geometric multiplicity algebraic multiplicity. Equality everywhere is required for the matrix to be diagonalizable.
92.4. Cayley–Hamilton
Every square matrix satisfies its own characteristic polynomial:
(substituting the matrix for — and interpreting the constant term as a multiple of ). This lets you express for any in terms of lower powers .