Linear algebra
Eigenvalues, Eigenvectors and Diagnolization
August 17, 2020
Question 1
Find the eigenvalues and eigenvectors of the following matrices.
a)
( 2 ?8 ?2 ?4
)
b)
? ?2 2 02 0 2 0 2 2
? ?
c)
? ????? 1 0 0 0 0 0 ?12 0 0 0 0 0 ?12 0 0 0 0 0 92 0 0 0 0 0 ?120
? ?????
Question 2 State whether the following are true or false. If false, explain why or give a counter-example.
a) Suppose T : R2[x] ?? R2[x] is a linear transformation with eigenvalues ?1 = 1,?2 = ?2,?3 = ?12. Then T is an isomorphism. b) A given eigenvector has only 1 eigenvalue associated to it. c) Suppose A is an n × n matrix, and ? is an eigenvalue for A. Then the columns of (A??In) are linearly independent. d) A given eigenvalue has only 1 eigenvector associated to it.
1
Question 3
Let B = (1,x,x2) be the standard basis for R2[x], and suppose
T : R2[x] ?? R2[x]
is a linear transformation whose matrix with respect to B is
AT,B =
? ? 5 2 ?46 3 ?5 10 4 ?8
? ?
We showed in class that this matrix has the following eigenvectors with as- sociated eigenvalues;
v1 =
? ?121
2
1
? ? with ?1 = ?1
v2 =
? ?121 1
? ? with ?2 = 1
v3 =
? ?231
3
1
? ? with ?3 = 0
a) Show that C = (v1,v2,v3) is a basis for R3. b) Let S = (e1,e2,e3) be the standard basis for R3. Find
PS??C (1) PC??S (2)
. c) Find the matrix multiplication
D = (PS??C)(AT,B)(PC??S)
d) What is the relationship of this matrix D with respect to the original transformation T?
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