Suppose that a 2x2 real matrix has the form http://www.cramster.com/Answer-Board...7037503255.gif. For which values of the numbers a and b will B be diagonalizable by a real matrix? Explain.

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- May 4th 2008, 10:31 AMmatty888Diagonal matrix
Suppose that a 2x2 real matrix has the form http://www.cramster.com/Answer-Board...7037503255.gif. For which values of the numbers a and b will B be diagonalizable by a real matrix? Explain.

- May 4th 2008, 12:19 PMTwistedOne151
Remember that when you diagonalize a matrix B, the diagonal values of the resulting diagonalized matrix will be the eigenvalues of B, and that a matrix P such that $\displaystyle P^{-1}BP$ is diagonal will have as it's columns the eigenvectors corresponding to those eigenvalues. Thus, we require that B have real eigenvectors. As our matrix is real, real eigenvalues are a necessary condition for real eigenvectors.

Note that $\displaystyle \det(I\lambda-B)=0$ gives $\displaystyle (\lambda-a)^2+b^2=0$.

This should point you in the right direction.

--Kevin C.