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Exercise 4

Avg difficulty 2.5

Consider the matrix A=[4−3−3−34−3−3−34]A = \begin{bmatrix} 4 & -3 & -3 \\ -3 & 4 & -3 \\ -3 & -3 & 4 \\ \end{bmatrix}.

  1. Find QQ and DD such that A=QDQTA=QDQ^T.

  2. Find the maximum and minimum values of q(u)=xTAxq(\uvec) = \xvec^TA\xvec among all unit vectors u\uvec.

  3. Describe the direction in which the minimum value occurs. What can you say about the direction in which the maximum occurs?

Austin, Understanding Linear Algebra, §7.2 Quadratic forms, Exercise 4

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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.