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/v, product below as a quadratic form:

2 In parts b and c of the preceding problem, the coefficient matrices are not symmetric with respect to the principal diagonal. Verify that by averaging the off-diagonal elements

_2 2l 5 31 and thus converting them, respectively, into ^ ^ 3 0 WS ®6t ^

same quadratic forms as before.

3 On the basis of their coefficient matrices (the symmetric versions;, determine by the determinantal test whether the quadratic forms in Exercise 11.3-la, b, and c are either positive definite or negative definite.

4 Express each quadratic form below as a matrix product involving a symmetric coefficient matrix:

(e) q = 3 u2 — 2utu2 + 4m,w3 + 5 u\ + 4 uj — 2u2u} (/) q = - u2 + 4uv - 6uw - 4tr - 7h-:

5 From the discriminants obtained from the symmetric coefficient matrices of the preceding problem, ascertain by the determinantal test which of the quadratic forms are positive definite and which are negative definite.

6 Find the characteristic roots of each of the following matrices:

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