Wednesday, March 26 − Lecture 32 : Eigenvalues of a matrix.
1. Definition of eigenvalue of a matrix.
2. Definition of the characteristic polynomial of a square matrix A.
3. Find the eigenvalues of a matrix.
4. Recognize some important properties of characteristic polynomials of a matrix A:
- The eigenvalues of a triangular matrix are the elements on its diagonal.
- The number λ = 0 is an eigenvalue of A if and only if det(A) = 0.
- The two matrices A and A have the same characteristic polynomial and hence
the same eigenvalues.
- Similar matrices have the same characteristic polynomial (C = B AB). -1
- The matrices AB and BA have the same characteristic polynomial.
32.1 Definitions − Let A be a square n × n matrix. Let λ be a variable over ℝ (the real
numbers). Then det(A − λI) is a polynomial in λ. (Verify this for 2 × 2 and 3 × 3
matrices.) .The expression det(A − λI) will always turnout to be a polynomial in λ, of the
n -1 n
det(A − λI) = a 0 a λ 1 ... + a n − 1 + λ .
Then p(λ) = det(A − λI) is called the characteristic polynomial of the matrix A. The
equation det(A − λI) = 0 is called the characteristic equation of A.
Note: Some authors define det(λI − A) = (−1) det(A − λI) as being the characteristic
polynomial. Since this essentially produces the same characteristic equationdet(A − λI)
= 0, we will not worry about this discrepancy.
Any root of the characteristic polynomial of A is called an eigenvalue of A. That is, if λ
is a solution of the polynomial equation
det(A − λI) = 0,
then λ 1 is an eigenvalue of A.
a) If A is the given 2 × 2 matrix, find all its eigenvalues.
det(A − λI) = (5 − λ) − 16
= (1 − λ)(9 −λ) = 0 The characteristic polynomial of Ais
(1 − λ)(9 −λ)
The characteristic equation of Ais
(1 − λ)(9 −λ) = 0
Thus λ =11 and λ = 92are two only eigenvalues of the matrix A. Note that the order in
which we present these eigenvalues is irrelevant.
b) If A is the matrix
then det(A − λI) = (2 − λ) [ (5 − λ) – 16] = (1 − λ)(2 − λ)(9 − λ) is characteristic
polynomial of A and (1 − λ)(2 − λ)(9 − λ) = 0 is characteristic equation ofA. So 1, 2
and 9 are the only three eigenvalues of A.
32.1.2 Remark − If A is a matrix such that det(A − λI) = (4 − λ) (1 − λ), then we see
that λ 1 4 is an eigenvalue and λ = 12is another eigenvalue. In this case we say that 4
is an eigenvalue of (algebraic) multiplicity 3.
Definition – If an eigenvalue is a root of the characteristic polynomial of multiplicity
m then we say it is an eigenvalue of algebraic multiplicity m.
32.1.3 Remark − If A is a matrix such that det(A − λI) = λ + 1, we see that A has no
eigenvalues which are real numbers. We can however accept "complex" roots such as
±√(−1) = ± i. If A is an n by n matrix and we view det(A − λI) as a polynomial over
the complex numbers ℂ, then there will always be n eigenvalues of A. Some of the
eigenvalues may be identical. This follows from a fundamental theorem in algebra
with says that any polynomial of degree n will have n complex roots.
32.1.4 Example − If A is the matrix
we verify, by ex