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Matrix diagonalization

  • Thread starter misterau
  • Start date
  • #1
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Homework Statement


A =

-10 6 3
-26 16 8
16 -10 -5

B =

0 -6 -16
0 17 45
0 -6 -16

(a) Show that 0, -1 and 2 are eigenvalues both of A and of B .
(b) Find invertible matrices P and Q so that (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)=

0 0 0
0 -1 0
0 0 2

(c) Find an invertible matrix R for which (R^-1)*(A)*(R) = B

Homework Equations





The Attempt at a Solution


I was able to do Q1 and Q2 but not Q3.
For Q2:
P =
0 1 1
-1 2 3
2 -1 -2

Q =
1 2 1
0 -5 -3
2 2 1

Not really sure about Q3, since matrix B is not in the form I am used too.
edit: I thought about it.
using, (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)
(Q)*(P^-1)*(A)*(P)*(Q^-1) = (Q)*(Q^-1)*(B)*(Q)*(Q^-1)
(Q)*(P^-1)*(A)*(P)*(Q^-1) = (B)
R = (P)*(Q^-1)
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,793
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Homework Statement


A =

-10 6 3
-26 16 8
16 -10 -5

B =

0 -6 -16
0 17 45
0 -6 -16

(a) Show that 0, -1 and 2 are eigenvalues both of A and of B .
(b) Find invertible matrices P and Q so that (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)=

0 0 0
0 -1 0
0 0 2

(c) Find an invertible matrix R for which (R^-1)*(A)*(R) = B

Homework Equations





The Attempt at a Solution


I was able to do Q1 and Q2 but not Q3.
For Q2:
P =
0 1 1
-1 2 3
2 -1 -2

Q =
1 2 1
0 -5 -3
2 2 1

Not really sure about Q3, since matrix B is not in the form I am used too.
edit: I thought about it.
using, (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)
(Q)*(P^-1)*(A)*(P)*(Q^-1) = (Q)*(Q^-1)*(B)*(Q)*(Q^-1)
(Q)*(P^-1)*(A)*(P)*(Q^-1) = (B)
R = (P)*(Q^-1)
Yes, of course! Now it's just a matter of finding Q^-1 and multiplying.
 

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