# Eigenvalue true and false

1. Dec 6, 2009

### shiri

Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.

A) 4 is an eigenvalue of A^T
B) 4 is an eigenvalue of (B^−1)AB
C) 265 is an eigenvalue of (A^4)+A+5I
D) 8 is an eigenvalue of A+(A^T)
E) 20 is an eigenvalue of AB
F) None of the above

I choose:
A)
B)

am I right?

Last edited: Dec 6, 2009
2. Dec 7, 2009

### omega101

Re: Eigenvalues

Let A and B be nn matrices, where B is invertible. Suppose that 5 is an eigenvalue of A, and 4 is an eigenvalue of B. Find ALL true statements below.
A. 20 is an eigenvalue of AB
B. 10 is an eigenvalue of A+AT
C. 34 is an eigenvalue of A2+A+4I
D. 5 is an eigenvalue of AT
E. 5 is an eigenvalue of B−1AB
F. None of the above

HELP!!!

3. Dec 8, 2009

### shiri

Re: Eigenvalues

anyone?

4. Dec 8, 2009

### Dick

Re: Eigenvalues

If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.

5. Dec 8, 2009

### shiri

Re: Eigenvalues

A) Say it was a 3x3 matrix. If I transpose the matrix, the eigenvalue still be 4.
B) BB^-1 becomes an identity and it lefts with A only.
C) A^4 + A + 5I = (4)^4 + 4 + 5 = 265. However, I'm not too sure about this statement.
D) Not too sure if I can simply add those two together. If I assume that upper and lower triangular part are zeros.
E) Obviously not a correct statement.

What do you think, Dick? A & B only?

I am very skeptical about C & D. Please help me on this, Dick.

6. Dec 8, 2009

### Dick

Re: Eigenvalues

Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?

7. Dec 8, 2009

### shiri

Re: Eigenvalues

A, B, C are true

D) If I assume there are non-zeros in the matrix

eg.
|1 4|+|1 3|=|2 7|
|3 2| |4 2| |7 4|

So, eval are roots of different roots and I have to use a quadratic formula to find those roots

E) Multiplying A&B will not come out as 20 b/c the matrix should be multiplied, not multiplying scalars (eigenvalues). So, False.

What do you think?

8. Dec 8, 2009

### Dick

Re: Eigenvalues

Yes, A, B and C are true. D and E are only true for some matrices, not for others. I'll give you examples where they aren't and you figure out why, ok? I can't really tell what you are trying to say for either of them. For D take A=[[4,1],[0,0]]. What are the eigenvalues of A, A^(T) and A+A^(T)? For E take A=[[4,0],[0,0]] and B=[[0,0],[0,5]]. What are the eigenvalues of A, B and AB? If you want another exercise, figure out specific matrices where D and E ARE true.
The text passage contains a series of questions related to eigenvalues of matrices, and the user is asking for help in determining the correct answers. The user also provides their own reasoning and answers for some of the questions.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Eigenvalue true and false

1. Dec 6, 2009

### shiri

Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.

A) 4 is an eigenvalue of A^T
B) 4 is an eigenvalue of (B^−1)AB
C) 265 is an eigenvalue of (A^4)+A+5I
D) 8 is an eigenvalue of A+(A^T)
E) 20 is an eigenvalue of AB
F) None of the above

I choose:
A)
B)

am I right?

Last edited: Dec 6, 2009
2. Dec 7, 2009

### omega101

Re: Eigenvalues

Let A and B be nn matrices, where B is invertible. Suppose that 5 is an eigenvalue of A, and 4 is an eigenvalue of B. Find ALL true statements below.
A. 20 is an eigenvalue of AB
B. 10 is an eigenvalue of A+AT
C. 34 is an eigenvalue of A2+A+4I
D. 5 is an eigenvalue of AT
E. 5 is an eigenvalue of B−1AB
F. None of the above

HELP!!!

3. Dec 8, 2009

### shiri

Re: Eigenvalues

anyone?

4. Dec 8, 2009

### Dick

Re: Eigenvalues

If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.

5. Dec 8, 2009

### shiri

Re: Eigenvalues

A) Say it was a 3x3 matrix. If I transpose the matrix, the eigenvalue still be 4.
B) BB^-1 becomes an identity and it lefts with A only.
C) A^4 + A + 5I = (4)^4 + 4 + 5 = 265. However, I'm not too sure about this statement.
D) Not too sure if I can simply add those two together. If I assume that upper and lower triangular part are zeros.
E) Obviously not a correct statement.

What do you think, Dick? A & B only?

I am very skeptical about C & D. Please help me on this, Dick.

6. Dec 8, 2009

### Dick

Re: Eigenvalues

Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?

7. Dec 8, 2009

### shiri

Re: Eigenvalues

A, B, C are true

D) If I assume there are non-zeros in the matrix

eg.
|1 4|+|1 3|=|2 7|
|3 2| |4 2| |7 4|

So, eval are roots of different roots and I have to use a quadratic formula to find those roots

E) Multiplying A&B will not come out as 20 b/c the matrix should be multiplied, not multiplying scalars (eigenvalues). So, False.

What do you think?

8. Dec 8, 2009

### Dick

Re: Eigenvalues

Yes, A, B and C are true. D and E are only true for some matrices, not for others. I'll give you examples where they aren't and you figure out why, ok? I can't really tell what you are trying to say for either of them. For D take A=[[4,1],[0,0]]. What are the eigenvalues of A, A^(T) and A+A^(T)? For E take A=[[4,0],[0,0]] and B=[[0,0],[0,5]]. What are the eigenvalues of A, B and AB? If you want another exercise, figure out specific matrices where D and E ARE true.
user
The text passage contains a series of questions related to eigenvalues of matrices, and the user is asking for help in determining the correct answers. The user also provides their own reasoning and answers for some of the questions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS