91Ó°ÊÓ

The \(n \times n\) matrix \(A\) has the property that, for every \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^{n},(\mathbf{u} A)(\mathbf{v} A)^{\mathbf{T}}=\mathbf{u v}^{\mathbf{T}}\). Show that \(A\) is orthogonal.

A basis \(\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\) for \(\mathbb{R}^{n}\) is orthonormal if \(\mathbf{e}_{i} \cdot \mathbf{e}_{i}=1(1 \leqslant i \leqslant n)\) and \(\mathbf{e}_{i} \cdot \mathbf{e}_{j}=0\) \((i \neq j)\). Prove that the linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is Euclidean if and only if \(T\) maps an orthonormal basis for \(\mathbb{R}^{n}\) to an orthonormal basis for \(\mathbb{R}^{n}\).

Let \(P=\left[\begin{array}{lll}u_{1} & v_{1} & w_{1} \\ u_{2} & v_{2} & w_{2} \\\ u_{3} & v_{3} & w_{3}\end{array}\right]\) be an orthogonal matrix such that \(P^{\mathrm{T}} A P\) \(=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)\) where \(\vec{A}\) is a \(3 \times 3\) real symmetric matrix. By the previous exercise, \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) are the eigenvalues of \(A\) with their proper multiplicities. Prove that \(\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right), \mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right), \mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)\) are corresponding unit eigenvectors.

The \(n \times n\) matrix \(A\) is said to be congruent to the \(n \times n\) matrix \(B\) if there is an \(n \times n\) orthogonal matrix \(P\) such that \(P^{\mathrm{T}} B P=A\). Prove that (a) every \(n \times n\) matrix is congruent to itself; (b) if \(A\) is congruent to \(B\), then \(B\) is congruent to \(A\); (c) if \(A, B, C\) are \(n \times n\) matrices such that \(A\) is congruent to \(B\), and \(B\) is congruent to \(C\), then \(A\) is congruent to \(C\).

Prove that a Euclidean linear transformation is associated with an orthogonal matrix with respect to any orthonormal basis for \(\mathbb{R}^{n}\).