Matrix diagonalization

In summary, the conversation involves finding eigenvalues for matrices A and B, finding invertible matrices P and Q such that (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q), and finding an invertible matrix R for which (R^-1)*(A)*(R) = B. The solution for Q3 is to use the given values of P and Q to find (Q^-1) and then multiply it by P to get R.
  • #1
20
0

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:
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  • #2
misterau said:

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.
 

1. What is matrix diagonalization?

Matrix diagonalization is a process in linear algebra where a square matrix is transformed into a diagonal matrix through a change of basis. This means that the new basis consists of eigenvectors of the original matrix, and the corresponding eigenvalues make up the diagonal entries of the diagonal matrix.

2. Why is matrix diagonalization important?

Matrix diagonalization is important because it allows us to simplify calculations and solve systems of linear equations more easily. It also helps in finding patterns and relationships within a matrix, and can be used to solve problems in various fields such as physics, engineering, and economics.

3. How is matrix diagonalization performed?

Matrix diagonalization is performed by finding the eigenvalues and eigenvectors of the original matrix. The eigenvectors are then used to form a matrix P, which is then multiplied by the diagonal matrix of eigenvalues and its inverse. This results in a diagonalized matrix.

4. What are the conditions for a matrix to be diagonalizable?

A matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the dimension of the matrix. Additionally, a matrix must have distinct eigenvalues in order to be diagonalizable. If a matrix does not meet these conditions, it is not diagonalizable.

5. Can any matrix be diagonalized?

No, not all matrices can be diagonalized. A matrix must meet certain conditions, such as having distinct eigenvalues and n linearly independent eigenvectors, in order to be diagonalizable. If these conditions are not met, the matrix cannot be diagonalized.

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