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Discover Linear Algebra

Jeremy Sylvestre

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Discovery guide 39.1 Discovery guide

Recall.

The standard inner product on \(\R^n\) (i.e. the dot product) can be calculated as a matrix product of column vectors:
\begin{equation*} \uvecinprod{u}{v} = \utrans{\uvec{v}} \uvec{u} \text{.} \end{equation*}
Similarly, the standard inner product on \(\C^n\) (i.e. the complex dot product) can be calculated as a matrix product of column vectors:
\begin{equation*} \uvecinprod{u}{v} = \adjoint{\uvec{v}} \uvec{u} \text{,} \end{equation*}
where \(\adjoint{\uvec{v}}\) means conjugate-transpose.
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Notation.

In this discovery guide we will write
\begin{align*} \amp {\inprod{\blank}{\blank}}_{\R} \text{,} \amp \amp {\inprod{\blank}{\blank}}_{\C} \end{align*}
to distinguish between the real and complex dot products, respectively.
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Discovery 39.1.

(a)

Fill in the blank with a matrix so that the formula below is valid for all vectors \(\uvec{u},\uvec{v}\) in \(\R^n\text{,}\) and would be true no matter what real \(n\times n\) matrix \(A\) is used:
\begin{equation*} {\inprod{\uvec{u}}{A \uvec{v}}}_{\R} = {\inprod{\fillinmath{XX} \uvec{u}}{\uvec{v}}}_{\R} \text{.} \end{equation*}
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Similarly, what matrix fills in the blank in the formula
\begin{equation*} {\inprod{A \uvec{u}}{\uvec{v}}}_{\R} = {\inprod{\uvec{u}}{\fillinmath{XX} \uvec{v}}}_{\R} \text{?} \end{equation*}
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(b)

Based on your answer to Task a, what property would a real matrix \(B\) need to possess in order for the formula
\begin{equation*} {\inprod{B \uvec{u}}{\uvec{v}}}_{\R} = {\inprod{\uvec{u}}{B \uvec{v}}}_{\R} \end{equation*}
to be true for all real vectors \(\uvec{u},\uvec{v}\text{?}\)
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(c)

Again, using your answer to Task a as a guide, what property would a real matrix \(C\) need to possess in order for the formula
\begin{gather} {\inprod{C \uvec{u}}{C \uvec{v}}}_{\R} = {\inprod{\uvec{u}}{\uvec{v}}}_{\R}\tag{✶} \end{gather}
to be true for all real vectors \(\uvec{u},\uvec{v}\text{?}\)
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Hint.
First apply the pattern you found Task a to the left-hand side of (✶). Then perhaps insert an identity matrix into the right-hand side, and re-evaluate the new equality in light of the pattern of Task a.
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Alternatively, you could turn both sides of (✶) into the matrix-multiplication version of the dot product, and investigate what happens when you take both of \(\uvec{u},\uvec{v}\) to be standard basis vectors.
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(d)

Fill in the blank with a matrix so that the formula below is valid for all vectors \(\uvec{u},\uvec{v}\) in \(\C^n\text{,}\) and would be true no matter what complex \(n\times n\) matrix \(A\) is used:
\begin{equation*} {\inprod{\uvec{u}}{A \uvec{v}}}_{\C} = {\inprod{\fillinmath{XX} \uvec{u}}{\uvec{v}}}_{\C} \text{.} \end{equation*}
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Similarly, what matrix fills in the blank in the formula
\begin{equation*} {\inprod{A \uvec{u}}{\uvec{v}}}_{\C} = {\inprod{\uvec{u}}{\fillinmath{XX} \uvec{v}}}_{\C} \text{?} \end{equation*}
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(e)

Based on your answer to Task d, what property would a complex matrix \(B\) need to possess in order for the formula
\begin{equation*} {\inprod{B \uvec{u}}{\uvec{v}}}_{\C} = {\inprod{\uvec{u}}{B \uvec{v}}}_{\C} \end{equation*}
to be true for all complex vectors \(\uvec{u},\uvec{v}\text{?}\)
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(f)

Again, using your answer to Task d as a guide, what property would a complex matrix \(C\) need to possess in order for the formula
\begin{equation*} {\inprod{C \uvec{u}}{C \uvec{v}}}_{\C} = {\inprod{\uvec{u}}{\uvec{v}}}_{\C} \end{equation*}
to be true for all complex vectors \(\uvec{u},\uvec{v}\text{?}\)
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Hint.
Employ the same strategy as for Task c. Or you probably could just infer the new complex pattern from the real pattern discovered in Task c.
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While we already have names for some of these things, they are examples of more abstract concepts, so we will use new terminology consistent with the abstract context.
(real) adjoint matrix
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the matrix that filled in the blank in Task a of Discovery 39.1
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(complex) adjoint matrix
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the matrix that filled in the blank in Task d of Discovery 39.1
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self-adjoint matrix
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a real matrix that satisfies the condition identified in Task b of Discovery 39.1 or a complex matrix that satisfies the condition identified in Task e of Discovery 39.1
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orthogonal matrix
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a real matrix that satisfies the condition identified in Task c of Discovery 39.1
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unitary matrix
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a complex matrix that satisfies the condition identified in Task f of Discovery 39.1
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Discovery 39.2.

What is the adjoint of the adjoint of a matrix \(A\text{?}\)
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Verify your conjecture against the inner product-based version of what adjoint means using the patterns of Task a of Discovery 39.1 (real case) and Task d of Discovery 39.1 (complex case).
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Our motivation for studying inner products was to connect algebra with geometry. Task b of Discovery 39.3 suggests a strong connection between an orthogonal matrix and the geometry of \(\R^n\text{.}\) Let’s explore that more through the inner product rather than through inspection of the matrix.
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Discovery 39.4. Orthogonal matrices: geometry.

A real matrix \(A\) is orthogonal if it satisfies
\begin{gather} {\inprod{A \uvec{u}}{A \uvec{v}}}_{\R} = {\inprod{\uvec{u}}{\uvec{v}}}_{\R}\tag{✶✶} \end{gather}
for every pair of vectors \(\uvec{u},\uvec{v}\) in \(\R^n\text{.}\)
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(b)

What if we apply rule (✶✶) with both \(\uvec{u},\uvec{v}\) to be the same vector \(\uvec{x}\text{?}\) What does
\begin{equation*} {\inprod{A \uvec{x}}{A \uvec{x}}}_{\R} = {\inprod{\uvec{x}}{\uvec{x}}}_{\R} \end{equation*}
say about about the geometry of how multiplication by \(A\) transforms vectors in \(\R^n\text{?}\)
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Summarize the pattern: Orthogonal \(A\) preserves .
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(c)

If we apply the previous pattern to the norm \(\norm{A \uvec{u} - A \uvec{v}}\text{,}\) we can also say:
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Orthogonal \(A\) preserves .
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(d)

Compare the angle between \(\uvec{u},\uvec{v}\) with the angle between \(A \uvec{u}, A \uvec{v}\text{:}\)
\begin{align*} \cos \theta_1 \amp = \frac{\uvecinprod{u}{v}}{\unorm{u} \unorm{v}} \text{,} \amp \cos \theta_2 \amp = \frac{\inprod{A \uvec{u}}{A \uvec{v}}}{\norm{A \uvec{u}} \norm{A \uvec{v}}}\text{.} \end{align*}
Combine two of the previous patterns into a new one:
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Orthogonal \(A\) preserves .
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(e)

What about when \(\theta = \pi/2\) ?
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Orthogonal \(A\) preserves .
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Discovery 39.6.

Suppose that \(A\) is an \(n \times n\) complex matrix and that \(W\) is an \(A\)-invariant subspace of \(\C^n\text{.}\) Verify that the orthogonal complement \(W^\perp\) is \(\adjoint{A}\)-invariant: assume \(\uvec{x}\) is orthogonal to every vector in \(W\) and verify that \(\adjoint{A} \uvec{x}\) is as well.
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Discovery 39.7.

Suppose \(V\) is a finite-dimensional complex inner product space, \(\basisfont{B} = \{ \uvec{v}_1, \uvec{v}_2, \dotsc, \uvec{v}_n \}\) is an orthonormal basis for \(V\text{,}\) and \(\basisfont{B'} = \{ \uvec{v}_1', \uvec{v}_2', \dotsc, \uvec{v}_n' \}\) is some other basis of \(V\text{.}\)
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(a)

What is the \((i,j)\) entry of the transition matrix \(\ucobmtrx{B'}{B}\text{?}\)
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Hint.
Columns of a transition matrix are coordinate vectors of “old” basis vectors relative to the “new” basis. Here the “new” basis \(\basisfont{B}\) is orthonormal, so the Expansion theorem (Theorem 37.5.5) might help.
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(b)

Use Task a to verify that if \(\basisfont{B'}\) is also orthonormal, then \(\ucobmtrx{B'}{B}\) is a unitary matrix.
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Hint.
What does the pattern of Task a say is the relationship between the entries of \(\ucobmtrx{B'}{B}\) and \(\ucobmtrx{B}{B'}\text{?}\)
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Also, recall that \(\uinvcobmtrx{B'}{B} = \ucobmtrx{B}{B'}\text{.}\)
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(c)

Explain how every unitary matrix can somehow be considered as a transition matrix between orthonormal bases of \(\C^n\text{.}\)
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