Linear Algebra

Seymour Lipschutz, Marc Lipson

Chapter 13

Linear Operators on Inner Product Spaces - all with Video Answers

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Chapter Questions

Problem 1

Find the adjoint of $F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ defined by
$$F(x, y, z)=(3 x+4 y-5 z, 2 x-6 y+7 z, 5 x-9 y+z)$$

Andrija Isakov

Andrija Isakov

Numerade Educator

Problem 2

Find the adjoint of $G: \mathbf{C}^{3} \rightarrow \mathbf{C}^{3}$ defined by
G(x, y, z)=[2 x+(1-i) y, \quad(3+2 i) x-4 i z, \quad 2 i x+(4-3 i) y-3 z]

Andrija Isakov

Numerade Educator

Problem 3

Prove Theorem 13.3: Let $\phi$ be a linear functional on an $n$ -dimensional inner product space $V$ Then there exists a unique vector $u \in V$ such that $\phi(v)=\langle v, u\rangle$ for every $v \in V$

ET

Ed Tam

Numerade Educator

Problem 4

Prove Theorem 13.1: Let $T$ be a linear operator on an $n$ -dimensional inner product space $V$. Then
(a) There exists a unique linear operator $T^{*}$ on $V$ such that
$$\langle T(u), v\rangle=\left\langle u, T^{*}(v)\right\rangle \quad \text { for all } u, v \in V$$
(b) Let $A$ be the matrix that represents $T$ relative to an orthonormal basis $S=\left\{u_{i}\right\} .$ Then the conjugate transpose $A^{*}$ of $A$ represents $T^{*}$ in the basis $S$

ET

Ed Tam

Numerade Educator

Problem 5

Prove Theorem 13.2:
(i) $\quad\left(T_{1}+T_{2}\right)^{*}=T_{1}^{*}+T_{2}^{*}$
(iii) $\quad\left(T_{1} T_{2}\right)^{*}=T_{2}^{*} T_{1}^{*}$
(ii) $\quad(k T)^{*}=\bar{k} T^{*}$
(iv) $\quad\left(T^{*}\right)^{*}=T$

ET

Ed Tam

Numerade Educator

Problem 6

Show that $(\mathrm{a}) \quad I^{*}=I,$ and $(\mathrm{b}) \quad 0^{*}=0$

Andrija Isakov

Numerade Educator

Problem 7

Suppose $T$ is invertible. Show that $\left(T^{-1}\right)^{*}=\left(T^{*}\right)^{-1}$

ET

Ed Tam

Numerade Educator

Problem 8

Let $T$ be a linear operator on $V$, and let $W$ be a $T$ -invariant subspace of $V$. Show that $W^{\perp}$ is invariant under $T^{*}$

ET

Ed Tam

Numerade Educator

Problem 9

Let $T$ be a linear operator on $V$. Show that each of the following conditions implies $T=0$ :
(i) $\langle T(u), v\rangle=0$ for every $u, v \in V$
(ii) $\quad V$ is a complex space, and $\langle T(u), u\rangle=0$ for every $u \in V$
(iii) $T$ is self-adjoint and $\langle T(u), u\rangle=0$ for every $u \in V$

ET

Ed Tam

Numerade Educator

Problem 10

Prove Theorem 13.6: The following conditions on an operator $U$ are equivalent:
(i) $U^{*}=U^{-1} ;$ that is, $U$ is unitary.
(ii) $\langle U(v), U(w)\rangle=\langle u, w\rangle$
(iii) $\|U(v)\|=\|v\|$

Andrew Misseldine

Andrew Misseldine

Numerade Educator

Problem 11

Let $U$ be a unitary (orthogonal) operator on $V$, and let $W$ be a subspace invariant under $U$. Show that $W^{\perp}$ is also invariant under $U$

ET

Ed Tam

Numerade Educator

Problem 12

Prove Theorem 13.9: The change-of-basis matrix from an orthonormal basis $\left\{u_{1}, \ldots, u_{n}\right\}$ into another orthonormal basis is unitary (orthogonal). Conversely, if $P=\left[a_{i j}\right]$ is a unitary (orthogonal) matrix, then the vectors $u_{i^{\prime}}=\sum_{j} a_{j i} u_{j}$ form an orthonormal basis.

Chris Trentman

Numerade Educator

Problem 13

Let $T$ be a symmetric operator. Show that (a) The characteristic polynomial $\Delta(t)$ of $T$ is a product of linear polynomials (over $\mathbf{R}$ ); (b) $T$ has a nonzero eigenvector.

Nick Johnson

Numerade Educator

Problem 14

Prove Theorem 13.11: Let $T$ be a symmetric operator on a real $n$ -dimensional inner product space $V$. Then there exists an orthonormal basis of $V$ consisting of eigenvectors of $T$. (Hence, $T$ can be represented by a diagonal matrix relative to an orthonormal basis.)

Nick Johnson

Numerade Educator

Problem 15

Let $q(x, y)=3 x^{2}-6 x y+11 y^{2} .$ Find an orthonormal change of coordinates (linear substitution) that diagonalizes the quadratic form $q$

Nick Johnson

Numerade Educator

Problem 16

Prove Theorem 13.12: Let $T$ be an orthogonal operator on a real inner product space $V$. Then there exists an orthonormal basis of $V$ in which $T$ is represented by a block diagonal matrix $M$ of the form
\[
M=\operatorname{diag}\left(1, \ldots, 1,-1, \ldots,-1,\left[\begin{array}{cc}
\cos \theta_{1} & -\sin \theta_{1} \\
\sin \theta_{1} & \cos \theta_{1}
\end{array}\right], \ldots,\left[\begin{array}{cc}
\cos \theta_{r} & -\sin \theta_{r} \\
\sin \theta_{r} & \cos \theta_{r}
\end{array}\right]\right)
\]

Nick Johnson

Numerade Educator

Problem 17

Determine which of the following matrices is normal:
(a) $A=\left[\begin{array}{ll}1 & i \\ 0 & 1\end{array}\right]$
(b) $\quad B=\left[\begin{array}{cc}1 & i \\ 1 & 2+i\end{array}\right]$

Andrija Isakov

Numerade Educator

Problem 18

Let $T$ be a normal operator. Prove the following:
(b) $\quad T-\lambda I$ is normal.
(a) $T(v)=0$ if and only if $T^{*}(v)=0$
(c) If $T(v)=\lambda v,$ then $T^{*}(v)=\bar{\lambda} v ;$ hence, any eigenvector of $T$ is also an eigenvector of $T^{*}$
(d) $\operatorname{If} T(v)=\lambda_{1} v$ and $T(w)=\lambda_{2} w$ where $\lambda_{1} \neq \lambda_{2},$ then $\langle v, w\rangle=0 ;$ that is, cigenvectors of $T$ belonging to distinct eigenvalues are orthogonal.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 20

Prove Theorem 13.14 : Let $T$ be any operator on a complex finite-dimensional inner product space $V .$ Then $T$ can be represented by a triangular matrix relative to an orthonormal basis of $V$

Nick Johnson

Numerade Educator

Problem 21

Prove Theorem $13.10 \mathrm{B}:$ The following are equivalent:
(i) $\quad P=T^{2}$ for some self-adjoint operator $T$
(ii) $\quad P=S^{*} S$ for some operator $S ;$ that is, $P$ is positive.
(iii) $P$ is self-adjoint and $\langle P(u), u\rangle \geq 0$ for every $u \in V$

Nick Johnson

Numerade Educator

Problem 22

Show that any operator $T$ is the sum of a self-adjoint operator and a skew-adjoint operator.

ET

Ed Tam

Numerade Educator

Problem 23

Prove: Let $T$ be an arbitrary linear operator on a finite-dimensional inner product space $V$. Then $T$ is a product of a unitary (orthogonal) operator $U$ and a unique positive operator $P$; that is, $T=U P .$ Furthermorc, if $T$ is invertible, then $U$ is also uniquely determined.

Nick Johnson

Numerade Educator

Problem 24

Let $V$ be the vector space of polynomials over $\mathbf{R}$ with inner product defined by
$$\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t$$
Give an example of a linear functional $\phi$ on $V$ for which Theorem 13.3 does not hold-that is, for which there is no polynomial $h(t)$ such that $\phi(f)=\langle f, h\rangle$ for every $f \in V$.

Nick Johnson

Numerade Educator

Problem 25

Find the adjoint of:
(a) $A=\left[\begin{array}{ll}5-2 i & 3+7 i \\ 4-6 i & 8+3 i\end{array}\right]$
(b) $\quad B=\left[\begin{array}{rr}3 & 5 i \\ i & -2 i\end{array}\right]$
(c) $C=\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]$

ET

Ed Tam

Numerade Educator

Problem 26

Let $T: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ be defined by $T(x, y, z)=(x+2 y, 3 x-4 z, y) .$ Find $T^{*}(x, y, z)$

Andrija Isakov

Numerade Educator

Problem 27

Let $T: \mathbf{C}^{3} \rightarrow \mathbf{C}^{3}$ be defined by $T(x, y, z)=[i x+(2+3 i) y, \quad 3 x+(3-i) z, \quad(2-5 i) y+i z]$ Find $T^{*}(x, y, z)$

Andrija Isakov

Numerade Educator

Problem 28

For each linear function $\phi$ on $V$, find $u \in V$ such that $\phi(v)=\langle v, u\rangle$ for every $v \in V$
(a) $\quad \phi: \mathbf{R}^{3} \rightarrow \mathbf{R}$ defined by $\phi(x, y, z)=x+2 y-3 z$
(b) $\phi: \mathbf{C}^{3} \rightarrow \mathbf{C}$ defined by $\phi(x, y, z)=i x+(2+3 i) y+(1-2 i) z$

Nick Johnson

Numerade Educator

Problem 29

Suppose $V$ has finite dimension. Prove that the image of $T^{*}$ is the orthogonal complement of the kernel of $T ;$ that is, $\operatorname{Im} T^{*}=(\text { Ker } T)^{\perp} .$ Hence, $\operatorname{rank}(T)=\operatorname{rank}\left(T^{*}\right)$

Nick Johnson

Numerade Educator

Problem 30

Show that $T^{*} T=0$ implies $T=0$

Scott Stetson

Numerade Educator

Problem 31

Let $V$ be the vector space of polynomials over $\mathbf{R}$ with inner product defined by $\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t .$ Let D be the derivative operator on $V ;$ that is, $\mathbf{D}(f)=d f / d t .$ Show that there is no operator $\mathbf{D}^{*}$ on $V$ such that $\langle\mathbf{D}(f), g\rangle=\left\langle f, \mathbf{D}^{*}(g)\right\rangle$ for every $f, g \in V .$ That is, $\mathbf{D}$ has no adjoint.

Nick Johnson

Numerade Educator

Problem 32

Unitary and Orthogonal Operators and Matrices
Find a unitary (orthogonal) matrix whose first row is
(a) $\quad(2 / \sqrt{13}, 3 / \sqrt{13})$
(b) a multiple of $(1,1-i)$
(c) a multiple of $(1,-i, 1-i)$

Nick Johnson

Numerade Educator

Problem 33

Prove that the products and inverses of orthogonal matrices are orthogonal. (Thus, the orthogonal matrices form a group under multiplication, called the orthogonal group.)

ET

Ed Tam

Numerade Educator

Problem 34

Prove that the products and inverses of unitary matrices are unitary. (Thus, the unitary matrices form a group under multiplication, called the unitary group.)

ET

Ed Tam

Numerade Educator

Problem 35

Show that if an orthogonal (unitary) matrix is triangular, then it is diagonal.

Mengchun Cai

Numerade Educator

Problem 36

Recall that the complex matrices $A$ and $B$ are unitarily equivalent if there exists a unitary matrix $P$ such that $B=P^{*} A P .$ Show that this relation is an equivalence relation.

Nick Johnson

Numerade Educator

Problem 37

Recall that the real matrices $A$ and $B$ are orthogonally equivalent if there exists an orthogonal matrix $P$ such that $B=P^{T} A P .$ Show that this relation is an cquivalence relation.

Hoan Nguyen

Numerade Educator

Problem 38

Let $W$ be a subspace of $V$. For any $v \in V$, let $v=w+w^{\prime}$, where $w \in W, w^{\prime} \in W^{\perp}$. (Such a sum is unique because $V=W \oplus W^{\perp}$.) Let $T: V \rightarrow V$ be defined by $T(v)=w-w^{\prime} .$ Show that $T$ is self-adjoint unitary operator on $V$

Nick Johnson

Numerade Educator

Problem 39

Let $V$ be an inner product space, and suppose $U: V \rightarrow V$ (not assumed linear) is surjective (onto) and preserves inner products; that is, $\langle U(v), U(w)\rangle=\langle u, w\rangle$ for every $v, w \in V .$ Prove that $U$ is linear and hence unitary.

Nick Johnson

Numerade Educator

Problem 40

Show that the sum of two positive (positive definite) operators is positive (positive definite).

ET

Ed Tam

Numerade Educator

Problem 41

Let $T$ be a linear operator on $V$ and $\operatorname{lct} f: V \times V \rightarrow K$ be defined by $f(u, v)=\langle T(u), v\rangle .$ Show that $f$ is an inner product on $V$ if and only if $T$ is positive definite.

Nick Johnson

Numerade Educator

Problem 42

Suppose $E$ is an orthogonal projection onto some subspace $W$ of $V$. Prove that $k I+E$ is positive (positive definite) if $k \geq 0(k>0)$

Nick Johnson

Numerade Educator

Problem 43

Consider the operator $T$ defined by $T\left(u_{i}\right)=\sqrt{\lambda_{i}} u_{i}, i=1, \ldots, n,$ in the proof of Theorem 13.10 A. Show that $T$ is positive and that it is the only positive operator for which $T^{2}=P$

Nick Johnson

Numerade Educator

Problem 44

Suppose $P$ is both positive and unitary. Prove that $P=I$

Nick Johnson

Numerade Educator

Problem 45

Detcrmine which of the following matrices are positive (positive definitc):
(i) $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right],(\text { ii })\left[\begin{array}{rr}0 & i \\ -i & 0\end{array}\right],$ (iii) $\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right],$ (iv) $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right],\left(\begin{array}{ll}v & 1 \\ 1 & 2\end{array}\right],$ (vi) $\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$

ET

Ed Tam

Numerade Educator

Problem 46

Prove that a $2 \times 2$ complex matrix $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is positive if and only if $(\mathrm{i}) A=A^{*},$ and (ii) $a, d$ and $|A|=a d-b c$ are nonnegative real numbers.

Elham Kordzadeh

Elham Kordzadeh

Numerade Educator

Problem 47

Prove that a diagonal matrix $A$ is positive (positive definite) if and only if every diagonal entry is a nonnegative (positive) real number.

ET

Ed Tam

Numerade Educator

Problem 48

For any operator $T,$ show that $T+T^{* *}$ is self-adjoint and $T-T^{*}$ is skew-adjoint

Hoan Nguyen

Numerade Educator

Problem 49

Suppose $T$ is self-adjoint. Show that $T^{2}(v)=0$ implies $T(v)=0 .$ Using this to prove that $T^{n}(v)=0$ also implics that $T(v)=0$ for $n>0$

Nick Johnson

Numerade Educator

Problem 50

Let $V$ be a complex inner product space. Suppose $\langle T(v), v\rangle$ is real for every $v \in V$. Show that $T$ is selfadjoint.

Nick Johnson

Numerade Educator

Problem 51

Suppose $T_{1}$ and $T_{2}$ are self-adjoint. Show that $T_{1} T_{2}$ is self-adjoint if and only if $T_{1}$ and $T_{1}$ commute; that is, $T_{1} T_{2}=T_{2} T_{1}$

ET

Ed Tam

Numerade Educator

Problem 52

For each of the following symmetric matrices $A,$ find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P^{\prime} A P$ is diagonal:
(a) $A=\left[\begin{array}{rr}1 & 2 \\ 2 & -2\end{array}\right]$
(b) $A=\left[\begin{array}{rr}5 & 4 \\ 4 & -1\end{array}\right]$
(c) $A=\left[\begin{array}{rr}7 & 3 \\ 3 & -1\end{array}\right]$

Nick Johnson

Numerade Educator

Problem 53

Find an orthogonal change of coordinates $X=P X^{\prime}$ that diagonalizes each of the following quadratic forms and find the corrcsponding diagonal quadratic form $q\left(x^{\prime}\right)$
(a) $q(x, y)=2 x^{2}-6 x y+10 y^{2}$
(b) $q(x, y)=x^{2}+8 x y-5 y^{2}$
(c) $q(x, y, z)=2 x^{2}-4 x y+5 y^{2}+2 x z-4 y z+2 z^{2}$

Victor Salazar

Numerade Educator

Problem 54

$\operatorname{Let} A=\left[\begin{array}{ll}2 & i \\ i & 2\end{array}\right] .$ Verify that $A$ is normal. Find a unitary matrix $P$ such that $P^{*} A P$ is diagonal. Find $P^{*} A P$

Nick Johnson

Numerade Educator

Problem 55

Show that a triangular matrix is normal if and only if it is diagonal.

Nick Johnson

Numerade Educator

Problem 56

Prove that if $T$ is normal on $V$, then $\|T(v)\|=\left\|T^{*}(v)\right\|$ for every $v \in V$. Prove that the converse holds in complex inner product spaces.

Nick Johnson

Numerade Educator

Problem 57

Show that self-adjoint, skew-adjoint, and unitary (orthogonal) operators are normal.

ET

Ed Tam

Numerade Educator

Problem 58

Suppose $T$ is normal. Prove that
(a) $T$ is self-adjoint if and only if its eigenvalues are real.
(b) $T$ is unitary if and only if its eigenvalues have absolute value 1
(c) $T$ is positive if and only if its eigenvalues are nonnegative real numbers.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 59

Show that if $T$ is normal, then $T$ and $T^{*}$ have the same kernel and the same image.

Nick Johnson

Numerade Educator

Problem 60

Suppose $T_{1}$ and $T_{2}$ are normal and commute. Show that $T_{1}+T_{2}$ and $T_{1} T_{2}$ are also normal.

Hoan Nguyen

Numerade Educator

Problem 61

Suppose $T_{1}$ is normal and commutes with $T_{2}$. Show that $T_{1}$ also commutes with $T_{2}^{\text {? }}$

Nick Johnson

Numerade Educator

Problem 63

Prove the following: Let $T_{1}$ and $T_{2}$ be normal operators on a complex finite-dimensional vector space $V$ Then there exists an orthonormal basis of $V$ consisting of cigenvectors of both $T_{1}$ and $T_{2}$. (That is, $T_{1}$ and $T_{2}$ can be simultaneously diagonalized.)

Nick Johnson

Numerade Educator

Problem 64

Show that inner product spaces $V$ and $W$ over $K$ are isomorphic if and only if $V$ and $W$ have the same dimension.

Nick Johnson

Numerade Educator

Problem 65

Show that inner product spaces $V$ and $W$ over $K$ are isomorphic if and only if $V$ and $W$ have the same dimension.

Hoan Nguyen

Numerade Educator

Problem 66

Let $V$ be an inner product space. Recall that each $u \in V$ determines a linear functional $\hat{u}$ in the dual space $V^{*}$ by the definition $\hat{u}(v)=\langle v, u\rangle$ for every $v \in V$. (See the text immediately preceding Theorem 13.3 .) Show that the map $u \mapsto \hat{u}$ is linear and nonsingular, and hence an isomorphism from $V$ onto $V^{*}$

ET

Ed Tam

Numerade Educator

Problem 67

Suppose $\left\{u_{1}, \ldots, u_{n}\right\}$ is an orthonormal basis of $V,$ Prove
(a) $\quad\left\langle a_{1} u_{1}+a_{2} u_{2}+\cdots+a_{n} u_{n}, \quad b_{1} u_{1}+b_{2} u_{2}+\cdots+b_{n} u_{n}\right\rangle=a_{1} \bar{b}_{1}+a_{2} \bar{b}_{2}+\ldots \bar{a}_{n} \bar{b}_{n}$
(b) Let $A=\left[a_{y}\right]$ be the matrix representing $T: V \rightarrow V$ in the basis $\left\{u_{i}\right\} .$ Then $a_{i j}=\left\langle T\left(u_{i}\right), u_{j}\right\rangle$

Victor Salazar

Numerade Educator

Problem 68

Show that there exists an orthonormal basis $\left\{u_{1}, \ldots, u_{n}\right\}$ of $V$ consisting of eigenvectors of $T$ if and only if there exist orthogonal projections $E_{1}, \ldots, E_{r}$ and scalars $\lambda_{1}, \ldots, \lambda_{r}$ such that
(i) $T=\lambda_{1} E_{1}+\cdots+\lambda_{r} E_{r}$
(ii) $\quad E_{1}+\cdots+E_{r}=I$
(iii) $\quad E_{i} E_{j}=0 \quad$ for $\quad i \neq j$

Nick Johnson

Numerade Educator

Problem 69

Suppose $V=U \oplus W$ and suppose $T_{1}: U \rightarrow V$ and $T_{2}: W \rightarrow V$ are linear. Show that $T=T_{1} \oplus T_{2}$ is also linear. Here $T$ is defined as follows: If $v \in V$ and $v=u+w$ where $u \in U, w \in W,$ then
$$T(v)=T_{1}(u)+T_{2}(w)$$

Hoan Nguyen

Numerade Educator