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 \(B\).

1. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3\); \[ T(a, b, c)=(a-2 b,-2 a+2 b+2 c, 2 b-c) \text {; }\nonumber \); dot product
2. \( \begin{array}{l} T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; \\ T\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{cc} c-a & d-b \\ a+2 c & b+2 d \end{array}\right] \end{array} \nonumber \)); inner product \(\left\langle\left[\begin{array}{cc}x & y \\ z & w\end{array}\right],\left[\begin{array}{cc}x^{\prime} & y^{\prime} \\ z^{\prime} & w^{\prime}\end{array}\right]\right\rangle=\) \(x x^{\prime}+y y^{\prime}+z z^{\prime}+w w^{\prime}\)
3. \( \begin{array}{l} T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=(b+c)+ \\ (a+c) x+(a+b) x^2 ; \text { inner product }\langle a+b x+ \\ \left.c x^2, a^{\prime}+b^{\prime} x+c^{\prime} x^2\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime} \end{array} \)

Answer

2. \(B=\left\{\left[\begin{array}{ll}
   1 & 0 \\
   0 & 0
   \end{array}\right],\left[\begin{array}{ll}
   0 & 1 \\
   0 & 0
   \end{array}\right],\left[\begin{array}{ll}
   0 & 0 \\
   1 & 0
   \end{array}\right],\left[\begin{array}{ll}
   0 & 0 \\
   0 & 1
   \end{array}\right]\right\}\) ; \(M_B(T)=\left[\begin{array}{rrrr}
   -1 & 0 & 1 & 0 \\
   0 & -1 & 0 & 1 \\
   1 & 0 & 2 & 0 \\
   0 & 1 & 0 & 2
   \end{array}\right]\)

Exercise \(\PageIndex{2}\)

Let \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by

\[ T(a, b)=(2 a+b, a-b) . \nonumber \nonumber \]

1. Show that \(T\) is symmetric if the dot product is used.
2. 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 \(\mathbb{R}^2\).

1. Show that \(T\) is symmetric.
2. 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:

1. The identity operator is symmetric.
2. \(r T\) is symmetric for all \(r\) in \(\mathbb{R}\).
3. \(S+T\) is symmetric.
4. If \(T\) is invertible, then \(T^{-1}\) is symmetric.
5. If \(S T=T S\), then \(S T\) is symmetric.

Answer

2. \(\langle\mathbf{v}, \quad(r T) \mathbf{w}\rangle=\langle\mathbf{v}, \quad r T(\mathbf{w})\rangle= r\langle\mathbf{v}, T(\mathbf{w})\rangle=r\langle T(\mathbf{v}), \mathbf{w}\rangle=\langle r T(\mathbf{v}), \mathbf{w}\rangle= \langle(r T)(\mathbf{v}), \mathbf{w}\rangle\)
4. Given \(\mathbf{v}\) and \(\mathbf{w}\), write \(T^{-1}(\mathbf{v})=\mathbf{v}_1\) and \(T^{-1}(\mathbf{w})=\mathbf{w}_1 . \quad\) Then \(\left\langle T^{-1}(\mathbf{v}), \mathbf{w}\right\rangle= \left\langle\mathbf{v}_1, T\left(\mathbf{w}_1\right)\right\rangle=\left\langle T\left(\mathbf{v}_1\right), \mathbf{w}_{|1\rangle}\right\rangle=\left\langle\mathbf{v}, T^{-1}(\mathbf{w})\right\rangle\).

Exercise \(\PageIndex{5}\)

In each case, show that \(T\) is symmetric and find an orthonormal basis of eigenvectors of \(T\).

1. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(a, b, c)=(2 a+2 c, 3 b, 2 a+\) \(5 c)\); use the dot product
2. \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 ; T(a, b, c)=(7 a-b,-a+\) \(7 b, 2 c)\); use the dot product
3. \(T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=3 b+(3 a+\) \(4 c) x+4 b x^2\); inner product \(\left\langle a+b x+c x^2, a^{\prime}+\right.\) \(\left.b^{\prime} x+c^{\prime} x^2\right\rangle=a a^{\prime}+b b^{\prime}+c c^{\prime}\)
4. \(T: \mathbf{P}_2 \rightarrow \mathbf{P}_2 ; T\left(a+b x+c x^2\right)=(c-a)+\) \(3 b x+(a-c) x^2 ;\) inner product as in part (c)

Answer

2. If \(B_0=\{(1,0,0),(0,1,0)\), \[ (0,0,1)\} \text {, then } M_{B_0}(T)=\left[\begin{array}{rrr} 7 & -1 & 0 \\ -1 & 7 & 0 \\ 0 & 0 & 2 \end{array}\right] \nonumber \] has an orthonormal basis of eigenvectors \(\left\{\frac{1}{\sqrt{2}}\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \frac{1}{\sqrt{2}}\left[\begin{array}{r}1 \\ -1 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]\right\} . \quad\) Hence an orthonormal basis of eigenvectors of \(T\) is \(\left\{\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{2}}(1,-1,0),(0,0,1)\right\}\).
3. If \(B_0=\left\{1, x, x^2\right\}\), then \(M_{B_0}(T)= \left[\begin{array}{rrr}-1 & 0 & 1 \\ 0 & 3 & 0 \\ 1 & 0 & -1\end{array}\right]\) has an orthonormal basis of eigenvectors

   \[
   \left\{\left[\begin{array}{l}
   0 \\
   1 \\
   0
   \end{array}\right], \frac{1}{\sqrt{2}}\left[\begin{array}{l}
   1 \\
   0 \\
   1
   \end{array}\right], \frac{1}{\sqrt{2}}\left[\begin{array}{r}
   1 \\
   0 \\
   -1
   \end{array}\right]\right\} .
   \nonumber \]

   Hence an orthonormal basis of eigenvectors of \(T\) is \(\left\{x, \frac{1}{\sqrt{2}}\left(1+x^2\right), \frac{1}{\sqrt{2}}\left(1-x^2\right)\right\}\).

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.

1. a. Show that \(T_A\) is symmetric if and only if \(P A=\) \(A^T P\).
2. 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.

1. Compute \(M_B(T)\), where \[ B=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} .\nonumber \nonumber \] Note the order!
2. Show that \(c_T(x)=\left[c_A(x)\right]^2\).
3. 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.

Answer

2. \(M_B(T)=\left[\begin{array}{cc} A & 0 \\ 0 & A \end{array}\right]\), so \(c_T(x)= \operatorname{det}\left[\begin{array}{cc} x I_2-A & 0 \\ 0 & x I_2-A
   \end{array}\right]=\left[c_A(x)\right]^2\)

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:

\[ \langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x} A \mathbf{y}^T, A=\left[\begin{array}{rr}
1 & -1 \\
-1 & 2
\end{array}\right]
\nonumber \nonumber \]

Exercise \(\PageIndex{9}\)

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

\[ M_{D B}(T)=\left[\frac{\left\langle\mathbf{d}_i, T\left(\mathbf{b}_j\right)\right\rangle}{\left\|\mathbf{d}_i\right\|^2}\right]
\nonumber \nonumber \]

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.

Answer

(1) ⇒ (2). If \(B=\left\{\mathbf{f}_1, \ldots, \mathbf{f}_n\right\}\) is an orthonormal basis of \(V\), then \(M_B(T)=\left[a_{i j}\right]\) where \(a_{i j}=\left\langle\mathbf{f}_i, T\left(\mathbf{f}_j\right)\right\rangle\) by Theorem 10.3.2. If (1) holds, then \(a_{j i}=\left\langle\mathbf{f}_j, T\left(\mathbf{f}_i\right)\right\rangle=-\left\langle T\left(\mathbf{f}_j\right), \mathbf{f}_i\right\rangle= -\left\langle\mathbf{f}_i, T\left(\mathbf{f}_j\right)\right\rangle=-a_{i j}\). Hence \(\left[M_V(T)\right]^T=-M_V(T)\), proving (2).

Exercise \(\PageIndex{13}\)

Let \(T: V \rightarrow V\) be a linear operator on an \(n\)-dimensional inner product space \(V\).

1. Show that \(T\) is symmetric if and only if it satisfies the following two conditions.
   1. \(c_T(x)\) factors completely over \(\mathbb{R}\).
   2. If \(U\) is a \(T\)-invariant subspace of \(V\), then \(U^{\perp}\) is also \(T\)-invariant.
2. 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 \(T^{\prime}(\mathbf{v})=\) \(\left\langle\mathbf{v}, T\left(\mathbf{f}_1\right)\right\rangle \mathbf{f}_1+\left\langle\mathbf{v}, T\left(\mathbf{f}_2\right)\right\rangle \mathbf{f}_2+\cdots+\left\langle\mathbf{v}, T\left(\mathbf{f}_n\right)\right\rangle \mathbf{f}_n=\) \(\sum_{i=1}^n\left\langle\mathbf{v}, T\left(\mathbf{f}_i\right)\right\rangle \mathbf{f}_i\)

1. Show that \((a T)^{\prime}=a T^{\prime}\).
2. Show that \((S+T)^{\prime}=S^{\prime}+T^{\prime}\).
3. Show that \(M_B\left(T^{\prime}\right)\) is the transpose of \(M_B(T)\).
4. Show that \(\left(T^{\prime}\right)^{\prime}=T\), using part (c). [Hint: \(M_B(S)=M_B(T)\) implies that \(S=T\).]
5. Show that \((S T)^{\prime}=T^{\prime} S^{\prime}\), using part (c).
6. Show that \(T\) is symmetric if and only if \(T=T^{\prime}\). [Hint: Use the expansion theorem and Theorem 10.3.3.]
7. Show that \(T+T^{\prime}\) and \(T T^{\prime}\) are symmetric, using parts (b) through (e).
8. 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 \(\left.i.\right]\)

Answer

3. The coefficients in the definition of \(T^{\prime}\left(\mathbf{f}_j\right)=\sum_{i=1}^n\left\langle\mathbf{f}_j, T\left(\mathbf{f}_i\right)\right\rangle \mathbf{f}_i\) are the entries in the \(j\) th column \(C_B\left[T^{\prime}\left(\mathbf{f}_j\right)\right]\) of \(M_B\left(T^{\prime}\right)\). Hence \(M_B\left(T^{\prime}\right)=\left[\left\langle\mathbf{f}_j, T\left(\mathbf{f}_j\right)\right\rangle\right]\), and this is the transpose of \(M_B(T)\) by Theorem 10.3.2.

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 \(\PageIndex{16}\)

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.

1. Show that \(\operatorname{proj}_U\) is a projection in the sense of Exercise 15.
2. 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.]