# Question: a write down a symmetric matrix a such that qx...

###### Question details

(a) Write down a symmetric matrix A such that Q(x) = xTAx.

(b) Find the eigenvalues and eigenvectors of A. (c) Find the maxima
and minima of Q(x) subject to ||x|| = 1. You should not use the
Lagrange method here.

(d) Find a vector w such that Q(w) = 16. Hint: The vector w could
be related to an eigenvector of A. It is possible to ﬁnd more than
one vectors.

The following exercises are for practice.

(e) (Hard) Graphically illustrate parts (c) and (d) following the
steps below:

(i) Consider the matrix M = (v1v2), where v1 and v2 are
eigenvectors of the matrix A. Use the change of variables, y = MTx,
and simplify the quadratic form Q. Hint: See relevant example in
the lecture slides. (ii) Draw two new axes, say Y1 and Y2 in the
direction of v1 and v2. Are these axes orthogonal to each other? On
this new set of axes draw contours Q(y) = c. (iii) You are now
ready to provide the required graphical illustrations.