# Linear Algebra eigenvalues help

• Oct 21st 2009, 07:14 PM
satweety12
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
• Oct 22nd 2009, 06:02 AM
HallsofIvy
Quote:

Originally Posted by satweety12
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.

Quote:

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?