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Math Help - Linear Algebra eigenvalues help

  1. #1
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    Linear Algebra eigenvalues help

    2 questions i need help with:

    1.Show that if a 6x6 matrix A has a negative determinant, then A has at least one positive eigenvalue (Hint: sketch the graph of the characteristic polynomial)

    2. Consider an eigenvalue lambda of an nxn matrix A. we know that lambda is an eignvalue of Atranspose as well (since A and Atranspose have the same characteristic polynomial). Compare the geometric multiplicities of lambda as an eignvalue of A and A transpose
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  2. #2
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    Quote Originally Posted by satweety12 View Post
    2 questions i need help with:

    1.Show that if a 6x6 matrix A has a negative determinant, then A has at least one positive eigenvalue (Hint: sketch the graph of the characteristic polynomial)
    Well, have you sketched the graph? Perhaps more simply, the characteristic polynomial will be of the form P(\lambda)= \lambda^6+ \cdot\cdot\cdot+ det(A)=. When \lambda= 0, P(0)= det(A)< 0. But for sufficiently large positive \lambda, P(\lambda)> 0.

    In fact, the same kind of argument shows that A has at least one postive eigenvalue also.

    As for the graph, just think of a parabola with y-intercept below the x-axis.

    2. Consider an eigenvalue lambda of an nxn matrix A. we know that lambda is an eignvalue of Atranspose as well (since A and Atranspose have the same characteristic polynomial). Compare the geometric multiplicities of lambda as an eignvalue of A and A transpose
    Yes, A and its transpose have the same eigenvalues. What can you say about the corresponding eigenvectors?

    Hint: \begin{bmatrix}7 & -3 \\ 10 & -4\end{bmatrix} has eigenvalues 1 and 2. What are its corresponding eigenvectors?

    Its transpose, \begin{bmatrix}7 & 10 \\ -3 & -4\end{bmatrix} also has eigenvalues 1 and 2. What are its corresponding eigenvectors?
    Last edited by HallsofIvy; October 22nd 2009 at 05:13 AM. Reason: Fixed some latex tags and quote tag.
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