Linear Algebra: Review Session 4
Eigenvalues and Eigenvectors
Computations and Diagonalizability
2 1 −2
Ex. 1. Let A be the matrix A = 0 1 0 .
1 1 −1
(a) Define what it means for a real number λ ∈ R to be an eigenvalue of A.
(b) Find all eigenvectors of A.
(c) Find a diagonal matrix D and an invertible matrix P such that A = P DP −1 , or show that no such
matrices exist.
2 0 −1
Ex. 2. Consider the matrix A = 3 0 −5.
1 0 0
(a) Find all eigenvalues of A.
(b) Find a diagonal matrix D and an invertible matrix P such that A = P DP −1 , or show that no such
matrices exist.
Ex. 3. Let A be the matrix
2
1
1
2
0 .
A= 0
−6 −1 −3
Find a diagonal matrix D and an invertible matrix P such that D = P AP −1 , or show that no such matrices
exist.
Ex. 4. Consider the matrix
1
A= 0
−2
3
Find a basis for R consisting of eigenvectors of A, or
Ex. 5. Consider the matrix
1
A = 2
1
0 −2
0 0 .
0 4
else prove that there is no such basis.
−2
5
2
0
0 .
2
Is A diagonalizable? Why or why not?
Proofs
Ex. 6. Let V be a finite-dimensional vector space, and let T : V → V be a linear transformation. Prove
that 0 is an eigenvalue of T if and only if the image (i.e., range) of T is not equal to V .
Ex. 7. Let A, B be n × n matrices that commute, i.e. AB = BA. Let v ∈ Rn be an eigenvector of A such
that Bv =
6 0. Prove that Bv is also an eigenvector of A.
Ex. 8. Suppose that a 3 × 3 matrix A has 0 as an eigenvalue.
(a) What are the possible values of the rank of A? Justify your answer.
(b) Let T : R3 → R3 be the linear transformation given by T (x) = Ax. Can T possibly be one-to-one? Can
T be onto? Justify your answer.
Ex. 9. Let A be an n × n matrix and α ∈ R be a scalar that is NOT an eigenvalue of A. Suppose that µ
is an eigenvalue for the matrix B = (A − αI)−1 with corresponding eigenvector v. Prove that v is also an
eigenvector for A and find a formula for the corresponding eigenvalue of A in terms of µ and α.
Ex. 10. Prove that if T : V → V is a linear map then the eigenspace E0 corresponding to the eigenvalue
λ = 0 is equal to Ker(T ).
Linear Algebra Review Session 4 Eigenvalues and Eigenvectors
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