Chapter 7.6, Problem 40E

Consider the matrix
$A=\left[\begin{array}{cccc}p& -q& -r& -s\\ q& p& s& -r\\ r& -s& p& q\\ s& r& -q& p\end{array}\right]$ ,
wherep, q, r, s are arbitrary real numbers. Compare this with Exercise 5.3.64.
a. Compute $A^T$A.
b. For which values of p, q, r, s is A invertible? Find the inverse if it exists.
c. Find the determinant of A.
d. Find the complex eigenvalues of A.
e. If $\stackrel{\to }{x}$ is a vector in ${ℝ}^4$ , what is the relationship between $\Vert \stackrel{\to }{x}\Vert$ and $\Vert A\stackrel{\to }{x}\Vert$ ?
f. Consider the numbers
$59=3^2+3^2+4^2+5^2$
and
$37=1^2+2^2+4^2+4^2$
Express the number
2183
as the sum of the squares of four integers:
$2183=a^2+b^2+c^2+d^2$

Hint: Part e is useful. Note that $2183=59\cdot 37$ .

g. The French mathematician Joseph−Louis Lagrange (1736−1813) showed that any prime number can be expressed as the sum of the squares of four integers. Using this fact and your work in part (f) as a guide, show that any positive integer can be expressed in this way.