ENSIA
Linear Algebra
Academic year 2021-2022
Second Academic Semester
Worksheet 5 : Eigenvalues and Eigenvectors
Exercise 1 Consider the matrices




1
3
3
2
4
3
A = −3 −5 −3 ; B = −4 −6 −3 .
3
3
1
3
3
1
1. Determine the eigenvalues of A and B.
2. Determine the subspace associated to each eigenvalue.
3. Explain why A is diagonalizable and B is not.
4. Give an invertible matrix P and a diagonal matrix D such that A = P DP −1 .
5. Compute An , n ∈ N.
Exercise 2 Explain without computing the eigenspace why the following matrix is not diagonalizable.


π 1 2
0 π 3
0 0 π
.
Exercise 3 True or False.
1. In finite dimension, an endomorphism admits a finite number of eigenvectors.
2. If A is diagonalizable, then A2 is diagonalizable.
3. If A2 is diagonalizable, then A is diagonalizable.
4. Any endomorphism of an R-vector space of odd dimension admits at least one eigenvalue.
5. The sum of two diagonalizable matrices is diagonalizable.
6. λ is an eigenvalue of A if, and only if, λ is an eigenvalue of AT .
7. A is diagonalizable if A = P −1 DP for some invertible matrix P and some diagonal
matrix D.
8. if A is diagonalizable then A is invertible.
9. If A is invertible, then A is diagonalizable.
10. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
11. To find an eigenvalue, reduce A to echelon form.
12. If v1 , v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
13. A and B are similar if and only if they have the same eigenvalues.
14. If A is 3 × 3 matrix with two eigenvalues, and each eigenspace is one-dimensional, then
A is diagonalizable.
1
Exercise 4 Let A = (aij )1⩽i,j⩽n such that aij = 1, ∀i, j ∈ {1, 2, · · · , n}.
Find the eigenvalues of A and their multiplicities.
Exercise 5 Consider the matrix


3 −11 4
A = −1 3 −1 .
−2 8 −3
1. Show, without computing the characteristic polynomial, that −1 is an eigenvalue of A.
2. Show that the matrix A is diagonalizable.
2