MATH136 Lecture Notes - Lecture 33: Algebraic Equation, Triangular Matrix, Invertible Matrix
14 May 2018
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Monday, July 17
−
Lecture 33 : Eigenvalues of a matrix. (Refers to 6.2)
Concepts:
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 λ1 = 0 is an eigenvalue of A if and only if det(A) = 0.
- The two matrices A and AT have the same characteristic polynomial and hence
the same eigenvalues.
- Similar matrices have the same characteristic polynomial (C = B-1AB).
- The matrices AB and BA have the same characteristic polynomial.
33.1 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
form
det(A
−
λI) = a0 + a1λ + ... + an − 1λn−1 + anλn.
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)ndet(A − λI) as being the characteristic
polynomial. Since this essentially produces the same characteristic equation det(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 λ1
is a solution of the polynomial equation
det(A
−
λI) = 0,
then λ1 is an eigenvalue of A.
33.1.1 Examples
a) If A is the given 2 × 2 matrix, find all its eigenvalues.
det(A
−
λI) = (5 − λ)2 − 16
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Document Summary
Monday, july 17 lecture 33 : eigenvalues of a matrix. (refers to 6. 2) Concepts: definition of eigenvalue of a matrix, definition of the characteristic polynomial of a square matrix a, find the eigenvalues of a matrix, 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 1 = 0 is an eigenvalue of a if and only if det(a) = 0. The two matrices a and at have the same characteristic polynomial and hence the same eigenvalues. Similar matrices have the same characteristic polynomial (c = b-1ab). The matrices ab and ba have the same characteristic polynomial. 33. 1 definitions let a be a square n n matrix. Let be a variable over (the real numbers). Then det(a i) is a polynomial in . (verify this for 2 2 and 3 3 matrices. )
