We primarily covered the homework due Saturday 4/6, 6.5 #13, #17, #18.
For #13, an important fact we uncovered is that if A and B are unitarily
equivalent matrices, then their norms are equal (proof below).
We also covered an old homework problem, 5.4 #18, as it pertains to showing that the inverse of any upper triangular matrix is also upper triangular. This is a very helpful fact to make the proof to #17 far simpler (proof below). We also sketched out how to prove #17 by induction.
We also showed that if A and B have the same eigenvalues, then they are similar.
Dr. Torok guided the discussion. Eric, Wendy, Taylor, Gabe, Terence and Ruth presented at the board. Thank you Taylor for all the pics.
SECT 6.5, PROB #13:
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| 6.5, #13 - A counter-example to the claim: If A and B are similar, then A and B are unitarily equivalent |
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| 6.5, #13 - Argument showing that if A and B are unitarily equivalent, then their norms are equal. |
SECT 6.5, PROB #17:
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| 6.5 #17 - a sketch of an induction argument. Induction Hypothesis: If A is an nxn unitary and upper triangular matrix, then A is diagonal. |
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| 6.5, #17 - Another argument using the fact that the inverse of an upper triangular matrix is also upper triangular |
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| 5.4, #18 - Showing that the inverse of a matrix is a linear combination of the powers of the original matrix. We find this specific combination by using Cayley-Hamilton |
SECT 6.5, PROB #19:
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| 6.5, #19 - Show that U is unitary |
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| 6.5, #19, show that U is self-adjoint |
OTHER TOPICS FOR SIMILAR MATRICES:
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| An argument showing that if A and B have the same eigenvalues, then they are similar. |
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| Some notes about similar matrices from Dr. Torok. |
