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Exercise 5

Avg difficulty 2.5

Consider the matrix B=[−214−22−1]B = \begin{bmatrix} -2 & 1 \\ 4 & -2 \\ 2 & -1 \\ \end{bmatrix}.

  1. Find the matrix AA so that q([x1x2])=∣Bx∣2=qA(x)q\left(\twovec{x_1}{x_2}\right) = \len{B\xvec}^2=q_A(\xvec).

  2. Find the maximum and minimum values of q(u)q(\uvec) among all unit vectors u\uvec and describe the directions in which they occur.

  3. What does the minimum value of q(u)q(\uvec) tell you about the matrix BB?

Austin, Understanding Linear Algebra, §7.2 Quadratic forms, Exercise 5

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.