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Chapter 8: Q7E (page 413)
The function is a quadratic form.
The given statement is FALSE.
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For each of the quadratic forms q listed in Exercises 1 through 3, find the matrix of q.
1.
a. Consider a complex upper triangularmatrix U with zeros on the diagonal. Show that u is nilpotent (i.e., thatlocalid="1659674833080" ). Compare with Exercises 3.3.78 and 3.3.79.
b. Consider a complexmatrix A that has zero as its only eigenvalue (with algebraic multiplicity n ). Use Exercise 45 to show that A is nilpotent.
Show that a quadratic formof two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix
Matrix is negative definite.
Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.
20.
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