7 - Y-K Lau's WebloQuestionsFiled under MATH1111— yklau...

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Unformatted text preview: Y.-K. Lau's WeblogNovember 29, 2010QuestionsFiled under: MATH1111— yklau @ 10:03 pm Below are some recently discussed questions.Question 1: Let and be two square matrices. If where , could we conclude is not similar to ? Ans. This is a typical mistake in logic. By definition, “is similar to ” means that one can find a nonsingular for which . Now one finds a pair with but it has NOimplication that one cannot find such . Indeed, we can easily find counterexample: Consider (the zero matrix). Then is certainly similar to , as well we can pick any pair so that Question 2(Text book: Chapter 6 Section 3 Qn 13 (7th edition) or Section 6.3 Qn 14 (8th edition)):Let be a diagonalizable matrix and let the diagonalizing matrix. Show that the column vectors of that correspond to nonzero eigenvalues of form a basis for the column space of . Ans. Write and Then . If , we have Let be the nonzero eigenvalues of . The above argument shows that ....
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7 - Y-K Lau's WebloQuestionsFiled under MATH1111— yklau...

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