Is Every Eigenvalue Claim Correct for Matrix Operations?

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In summary, an eigenvalue is a characteristic value of a matrix or transformation that represents how it stretches or compresses a vector. To calculate eigenvalues, one can take the determinant of the matrix and solve for the values that make it equal to zero. Eigenvalues are significant in understanding the behavior of a matrix or transformation, such as determining if it is invertible or if it has special properties. They can also be complex numbers if the matrix or transformation has complex entries. The main difference between true and false eigenvalues is that true eigenvalues satisfy the equation for eigenvalue calculation and have corresponding eigenvectors, while false eigenvalues do not.
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shiri
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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?
 
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  • #2


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


shiri said:
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?

anyone?
 
  • #4


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


Dick said:
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.

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


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


Dick said:
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?

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


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.
 

Related to Is Every Eigenvalue Claim Correct for Matrix Operations?

1. What is an eigenvalue?

An eigenvalue is a number that represents how a particular matrix or transformation stretches or compresses a vector during a transformation. It is a characteristic value of the matrix or transformation.

2. How do you calculate eigenvalues?

Eigenvalues can be calculated by taking the determinant of the matrix and solving for the values that make the determinant equal to zero. These values are the eigenvalues.

3. What is the significance of eigenvalues?

Eigenvalues are used to understand the behavior of a matrix or transformation. They can determine if the matrix is invertible, if the transformation stretches or compresses a vector, and if the matrix or transformation has special properties.

4. Can eigenvalues be complex numbers?

Yes, eigenvalues can be complex numbers. In fact, if the matrix or transformation has complex entries, it is likely that the eigenvalues will also be complex.

5. What is the difference between true and false eigenvalues?

True eigenvalues are values that satisfy the equation for the eigenvalue calculation, while false eigenvalues are values that do not satisfy the equation. False eigenvalues do not have corresponding eigenvectors, while true eigenvalues do.

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