Chapter 8

If \(A\) is square and real, show that \(A=0\) if and only if every eigenvalue of \(A^{T} A\) is \(0 .\)

If \(A \sim B\) show that \(A\) is invertible if and only if \(B\) is invertible.

Show that a matrix \(N\) is normal if and only if \(\bar{N} N^{T}=N^{T} \bar{N}\).

If \(A\) is an \(n \times n\) positive definite matrix and \(U\) is an \(n \times m\) matrix of rank \(m\), show that \(U^{T} A U\) is positive definite.

If \(A\) is positive definite, show that each diagonal entry is positive.

Consider \(A=\left[\begin{array}{lll}0 & 0 & a \\ 0 & b & 0 \\ a & 0 & 0\end{array}\right] .\) Show that \(c_{A}(x)=(x-b)(x-a)(x+a)\) and find an orthogonal matrix \(P\) such that \(P^{-1} A P\) is diagonal.

Consider the linear system \(3 x+y+4 z=3\). In each case solve the system by \(4 x+3 y+z=1\) reducing the augmented matrix to reduced row-echelon form over the given field: a. \(\mathbb{Z}_{5}\) b. \(\mathbb{Z}_{7}\)

Given a SVD for an invertible matrix A, find one for \(A^{-1}\). How are \(\Sigma_{A}\) and \(\Sigma_{A^{-1}}\) related?

If \(\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)^{T}\) is a column of variables, \(A=A^{T}\) is \(n \times n, B\) is \(1 \times n,\) and \(c\) is a constant, \(\mathbf{x}^{T} A \mathbf{x}+B \mathbf{x}=c\) is called a quadratic equation in the variables \(x_{i}\) a. Show that new variables \(y_{1}, \ldots, y_{n}\) can be found such that the equation takes the form $$\lambda_{1} y_{1}^{2}+\cdots+\lambda_{r} y_{r}^{2}+k_{1} y_{1}+\cdots+k_{n} y_{n}=c$$ b. Put \(x_{1}^{2}+3 x_{2}^{2}+3 x_{3}^{2}+4 x_{1} x_{2}-4 x_{1} x_{3}+5 x_{1}-6 x_{3}=7\) in this form and find variables \(y_{1}, y_{2}, y_{3}\) as in (a).

Let \(U\) be a subspace of \(\mathbb{R}^{n} .\) If \(\mathbf{x}\) in \(\mathbb{R}^{n}\) can be written in any way at all as \(\mathbf{x}=\mathbf{p}+\mathbf{q}\) with \(\mathbf{p}\) in \(U\) and \(\mathbf{q}\) in \(U^{\perp}\), show that necessarily \(\mathbf{p}=\operatorname{proj}_{U} \mathbf{x}\).