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Exercise 2

Avg difficulty 2.5

Consider the quadratic form q([x1x2])=3x12−4x1x2+6x22.q\left(\twovec{x_1}{x_2}\right) = 3x_1^2 - 4x_1x_2 + 6x_2^2.

  1. Find a matrix AA such that q(x)=xTAxq(\xvec) = \xvec^TA\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.

Austin, Understanding Linear Algebra, §7.2 Quadratic forms, Exercise 2

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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.