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# sol67 - Partial Solution Set Leon 6.7 6.7.1 For each of the...

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Partial Solution Set, Leon § 6.7 6.7.1 For each of the following, compute the determinants of all of the leading principal submatrices and use them to determine whether the matrix is positive deﬁnite. (In other words, compute the leading principal minors.) 1. A = ± 2 - 1 - 1 2 ² . Leading principal minors are 2 and 3; positive deﬁnite. 2. A = ± 3 4 4 3 ² . Leading principal minors are 3, - 7; not positive deﬁnite. 3. A = 6 4 - 2 4 5 3 - 2 3 6 . Leading principal minors are 6, 14, - 38; not positive deﬁnite. 4. A = 4 2 1 2 3 - 2 1 - 2 5 . Leading principal minors are 4, 8, and 13; positive deﬁnite. 6.7.5 For each of the following, ﬁnd the Cholesky decomposition A = BB T , where B is a lower triangular matrix. (Problem 6.7.4 is to ﬁnd factorizations of the form LDL T , which is an intermediate step here.) 1. A = ± 4 2 2 10 ² . (d) A = 9 3 - 6 3 4 1 - 6 1 9 .

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sol67 - Partial Solution Set Leon 6.7 6.7.1 For each of the...

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