10.3E: Orthogonal Diagonalization Exercises

Exercise \(\PageIndex{1}\) In each case, show that \(T\) is symmetric by calculating \(M_B(T)\) for some orthonormal basis \(B\).
a. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3\);
\[
T(a, b, c)=(a-2 b,-2 a+2 b+2 c, 2 b-c) \text {; }
\]
dot product
b.
\[
\begin{array}{l}
T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; \\
T\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]=\left[\begin{array}{cc}
c-a & d-b \\
a+2 c & b+2 d
\end{array}\right]
\end{array}
\]
inner product \(\left\langle\left[\begin{array}{cc}x & y \\ z & w\end{array}\right],\left[\begin{array}{cc}x^{\prime} & y^{\prime} \\ z^{\prime} & w^{\prime}\end{array}\right]\right\rangle=\) \(x x^{\prime}+y y^{\prime}+z z^{\prime}+w w^{\prime}\)
c.
\[
\begin{array}{l}
T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=(b+c)+ \\
(a+c) x+(a+b) x^2 ; \text { inner product }\langle a+b x+ \\
\left.c x^2, a^{\prime}+b^{\prime} x+c^{\prime} x^2\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime}
\end{array}
\]Exercise \(\PageIndex{2}\) Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by
\[
T(a, b)=(2 a+b, a-b) .
\]
a. Show that \(T\) is symmetric if the dot product is used.
b. Show that \(T\) is not symmetric if \(\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x} A \mathbf{y}^T\), where \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]\). [Hint: Check that \(B=\) \(\{(1,0),(1,-1)\}\) is an orthonormal basis.]

Exercise \(\PageIndex{3}\) Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by \(T(a\), \(b)=(a-b, b-a)\). Use the dot product in \(\mathbb{R}^2\).
a. Show that \(T\) is symmetric.

b. Show that \(M_B(T)\) is not symmetric if the orthogonal basis \(B=\{(1,0),(0,2)\}\) is used. Why does this not contradict Theorem 10.3.3?

Exercise \(\PageIndex{4}\) Let \(V\) be an \(n\)-dimensional inner product space, and let \(T\) and \(S\) denote symmetric linear operators on \(V\). Show that:
a. The identity operator is symmetric.
b. \(r T\) is symmetric for all \(r\) in \(\mathbb{R}\).
c. \(S+T\) is symmetric.
d. If \(T\) is invertible, then \(T^{-1}\) is symmetric.
e. If \(S T=T S\), then \(S T\) is symmetric.

Exercise \(\PageIndex{5}\) In each case, show that \(T\) is symmetric and find an orthonormal basis of eigenvectors of \(T\).
a. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(a, b, c)=(2 a+2 c, 3 b, 2 a+\) \(5 c)\); use the dot product
b. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(a, b, c)=(7 a-b,-a+\) \(7 b, 2 c)\); use the dot product
c. \(T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=3 b+(3 a+\) \(4 c) x+4 b x^2\); inner product \(\left\langle a+b x+c x^2, a^{\prime}+\right.\) \(\left.b^{\prime} x+c^{\prime} x^2\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime}\)
d. \(T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=(c-a)+\) \(3 b x+(a-c) x^2 ;\) inner product as in part (c)

Exercise \(\PageIndex{6}\) If \(A\) is any \(n \times n\) matrix, let \(T_A\) : \(\mathbb{R}^n \rightarrow \mathbb{R}^n\) be given by \(T_A(\mathbf{x})=A \mathbf{x}\). Suppose an inner product on \(\mathbb{R}^n\) is given by \(\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T P \mathbf{y}\), where \(P\) is a positive definite matrix.
a. Show that \(T_A\) is symmetric if and only if \(P A=\) \(A^T P\).
b. Use part (a) to deduce Example 10.3.3.

Exercise \(\PageIndex{7}\) Let \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) be given by \(T(X)=A X\), where \(A\) is a fixed \(2 \times 2\) matrix.
a. Compute \(M_B(T)\), where
\[
B=\left\{\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right],\left[\begin{array}{ll}
0 & 0 \\
1 & 0
\end{array}\right],\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right],\left[\begin{array}{ll}
0 & 0 \\
0 & 1
\end{array}\right]\right\} .
\]
Note the order!
b. Show that \(c_T(x)=\left[c_A(x)\right]^2\).
c. If the inner product on \(\mathbf{M}_{22}\) is \(\langle X, Y\rangle=\) \(\operatorname{tr}\left(X Y^T\right)\), show that \(T\) is symmetric if and only if \(A\) is a symmetric matrix.

Exercise \(\PageIndex{8}\) Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by \(T(a\), \(b)=(b-a, a+2 b)\). Show that \(T\) is symmetric if the dot product is used in \(\mathbb{R}^2\) but that it is not symmetric if the following inner product is used:
\[
\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x} A \mathbf{y}^T, A=\left[\begin{array}{rr}
1 & -1 \\
-1 & 2
\end{array}\right]
\]
Exercise \(\PageIndex{9}\) If \(T: V \rightarrow V\) is symmetric, write \(T^{-1}(W)=\{\mathbf{v} \mid T(\mathbf{v})\) is in \(W\}\). Show that \(T(U)^{\perp}=\) \(T^{-1}\left(U^{\perp}\right)\) holds for every subspace \(U\) of \(V\).

Exercise \(\PageIndex{10}\) Let \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) be defined by \(T(X)=P X Q\), where \(P\) and \(Q\) are nonzero \(2 \times 2\) matrices. Use the inner product \(\langle X, Y\rangle=\) \(\operatorname{tr}\left(X Y^T\right)\). Show that \(T\) is symmetric if and only if either \(P\) and \(Q\) are both symmetric or both are scalar multiples of \(\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\). [Hint: If \(B\) is as in part (a) of Exercise 7, then \(M_B(T)=\left[\begin{array}{ll}a P & c P \\ b P & d P\end{array}\right]\)

in block form, where \(Q=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\). If \(B_0=\) \(\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\}\), then \(M_B(T)=\left[\begin{array}{cc}p Q^T & q Q^T \\ r Q^T & s Q^T\end{array}\right]\), where \(P=\left[\begin{array}{cc}p & q \\ r & s\end{array}\right]\). Use the fact that \(c P=b P^T \Rightarrow\left(c^2-b^2\right) P=0\).]

Exercise \(\PageIndex{11}\) Let \(T: V \rightarrow W\) be any linear transformation and let \(B=\left\{\mathbf{b}_1, \ldots, \mathbf{b}_n\right\}\) and \(D=\) \(\left\{\mathbf{d}_1, \ldots, \mathbf{d}_m\right\}\) be bases of \(V\) and \(W\), respectively. If \(W\) is an inner product space and \(D\) is orthogonal, show that
\[
M_{D B}(T)=\left[\frac{\left\langle\mathbf{d}_i, T\left(\mathbf{b}_j\right)\right\rangle}{\left\|\mathbf{d}_i\right\|^2}\right]
\]
This is a generalization of Theorem 10.3.2.

Exercise \(\PageIndex{12}\) Let \(T: V \rightarrow V\) be a linear operator on an inner product space \(V\) of finite dimension. Show that the following are equivalent.
1. \(\langle\mathbf{v}, T(\mathbf{w})\rangle=-\langle T(\mathbf{v}), \mathbf{w}\rangle\) for all \(\mathbf{v}\) and \(\mathbf{w}\) in \(V\).
2. \(M_B(T)\) is skew-symmetric for every orthonormal basis \(B\).
3. \(M_B(T)\) is skew-symmetric for some orthonormal basis \(B\).

Such operators \(T\) are called skew-symmetric operators.

Exercise \(\PageIndex{13}\) Let \(T: V \rightarrow V\) be a linear operator on an \(n\)-dimensional inner product space \(V\).
a. Show that \(T\) is symmetric if and only if it satisfies the following two conditions.
i. \(c_T(x)\) factors completely over \(\mathbb{R}\).
ii. If \(U\) is a \(T\)-invariant subspace of \(V\), then \(U^{\perp}\) is also \(T\)-invariant.

b. Using the standard inner product on \(\mathbb{R}^2\), show that \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) with \(T(a, b)=(a, a+b)\) satisfies condition (i) and that \(S: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) with \(S(a, b)=(b,-a)\) satisfies condition (ii), but that neither is symmetric. (Example 9.3.4 is useful for \(S\).)
[Hint for part (a): If conditions (i) and (ii) hold, proceed by induction on \(n\). By condition (i), let \(\mathbf{e}_1\) be an eigenvector of \(T\). If \(U=\) \(\mathbb{R} \mathbf{e}_1\), then \(U^{\perp}\) is \(T\)-invariant by condition (ii), so show that the restriction of \(T\) to \(U^{\perp}\) satisfies conditions (i) and (ii). (Theorem 9.3.1 is helpful for part (i)). Then apply induction to show that \(V\) has an orthogonal basis of eigenvectors (as in Theorem 10.3.6)].

Exercise \(\PageIndex{14}\) Let \(B=\left\{\mathbf{f}_1, \mathbf{f}_2, \ldots, \mathbf{f}_n\right\}\) be an orthonormal basis of an inner product space \(V\). Given \(T: V \rightarrow V\), define \(T^{\prime}: V \rightarrow V\) by \(T^{\prime}(\mathbf{v})=\) \(\left\langle\mathbf{v}, T\left(\mathbf{f}_1\right)\right\rangle \mathbf{f}_1+\left\langle\mathbf{v}, T\left(\mathbf{f}_2\right)\right\rangle \mathbf{f}_2+\cdots+\left\langle\mathbf{v}, T\left(\mathbf{f}_n\right)\right\rangle \mathbf{f}_n=\) \(\sum_{i=1}^n\left\langle\mathbf{v}, T\left(\mathbf{f}_i\right)\right\rangle \mathbf{f}_i\)
a. Show that \((a T)^{\prime}=a T^{\prime}\).
b. Show that \((S+T)^{\prime}=S^{\prime}+T^{\prime}\).
c. Show that \(M_B\left(T^{\prime}\right)\) is the transpose of \(M_B(T)\).
d. Show that \(\left(T^{\prime}\right)^{\prime}=T\), using part (c). [Hint: \(M_B(S)=M_B(T)\) implies that \(S=T\).]
e. Show that \((S T)^{\prime}=T^{\prime} S^{\prime}\), using part (c).
f. Show that \(T\) is symmetric if and only if \(T=T^{\prime}\). [Hint: Use the expansion theorem and Theorem 10.3.3.]
g. Show that \(T+T^{\prime}\) and \(T T^{\prime}\) are symmetric, using parts (b) through (e).

h. Show that \(T^{\prime}(\mathbf{v})\) is independent of the choice of orthonormal basis B. [Hint: If \(D=\left\{\mathbf{g}_1\right.\), \(\left.\ldots, \mathbf{g}_n\right\}\) is also orthonormal, use the fact that \(\mathbf{f}_i=\sum_{j=1}^n\left\langle\mathbf{f}_i, \mathbf{g}_j\right\rangle \mathbf{g}_j\) for each \(\left.i.\right]\)

Exercise \(\PageIndex{15}\) Let \(V\) be a finite dimensional inner product space. Show that the following conditions are equivalent for a linear operator \(T: V \rightarrow\) \(V\).
1. \(T\) is symmetric and \(T^2=T\).
2. \(M_B(T)=\left[\begin{array}{cc}I_r & 0 \\ 0 & 0\end{array}\right]\) for some orthonormal basis \(B\) of \(V\).

An operator is called a projection if it satisfies these conditions. [Hint: If \(T^2=T\) and \(T(\mathbf{v})=\lambda \mathbf{v}\), apply \(T\) to get \(\lambda \mathbf{v}=\lambda^2 \mathbf{v}\). Hence show that 0,1 are the only eigenvalues of \(T\).]
Exercise 10.3.16 Let \(V\) denote a finite dimensional inner product space. Given a subspace \(U\), define \(\operatorname{proj}_U: V \rightarrow V\) as in Theorem 10.2.7.
a. Show that \(\operatorname{proj}_U\) is a projection in the sense of Exercise 15.
b. If \(T\) is any projection, show that \(T=\operatorname{proj}_U\), where \(U=\operatorname{im} T\). [Hint: Use \(T^2=T\) to show that \(V=\operatorname{im} T \oplus \operatorname{ker} T\) and \(T(\mathbf{u})=\mathbf{u}\) for all \(\mathbf{u}\) in \(\operatorname{im} T\). Use the fact that \(T\) is symmetric to show that \(\operatorname{ker} T \subseteq(\operatorname{im} T)^{\perp}\) and hence that these are equal because they have the same dimension.]