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Unformatted text preview: at M satisfies M T ( AB BT AT )T BT AT AB M , i. e. M=‐MT. In the solution to the Problem Set 2, I had written that such a matrix is called a skew symmetric matrix as it is the opposite of a symmetric matrix. We are told that is the real eigenvalue of M. Let w be a corresponding eigenvector. Then w 0 and Mw w From here, we need to somehow figure out what l must be. Let’s follow the ideas of the Problem Set 2 solution and also the discussion in class. By multiplying on the left by wT, we get wT Mw wT w wT w. Let us see what we can say about wT Mw. First, we note that wT Mw is a scalar and therefore equals its own transpose. Therefore, wT Mw ( wT Mw) T wT M T w wT Mw. So, it must be the case that wT Mw =0. Therefore, wT w 0 But w 0 and therefor wT w w 0,
2 0 Thus, if is the real eigenvalue of matrix M then it must be 0. In general, a skew symmetric matrix has either purely imaginary or zero eigenvalues. Notice the beautiful structure ‐‐‐ for a symmetric matrix, eigenvalues are always real; for a skew‐symmetric matrix they are always purely imaginary!!...
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.
 Fall '10
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