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Eigenvalue true and false

  1. Dec 6, 2009 #1
    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. jcsd
  3. Dec 7, 2009 #2
    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!!!
     
  4. Dec 8, 2009 #3
    Re: Eigenvalues

    anyone?
     
  5. Dec 8, 2009 #4

    Dick

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    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.
     
  6. Dec 8, 2009 #5
    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.
     
  7. Dec 8, 2009 #6

    Dick

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    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?
     
  8. Dec 8, 2009 #7
    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?
     
  9. Dec 8, 2009 #8

    Dick

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    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.
     
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