Section 36.6 Theory
Subsection 36.6.1 Properties of inner products
Here we list algebraic properties of inner products, including those prescribed by the inner product axioms.
Proposition 36.6.1. Algebra rules of real inner products.
Suppose \(\inprod{\blank}{\blank}\) is a real inner product. Then the following are true for all vectors \(\uvec{u}\text{,}\) \(\uvec{v}\text{,}\) and \(\uvec{w}\) in the associated inner product space, and all scalars \(k\text{.}\)
-
\(\uvecinprod{v}{u} = \uvecinprod{u}{v}\text{.}\)
-
\(\inprod{\uvec{u}}{\uvec{v} + \uvec{w}} = \uvecinprod{u}{v} + \uvecinprod{u}{w}\text{.}\)
-
\(\inprod{\uvec{u}}{\uvec{v} - \uvec{w}} = \uvecinprod{u}{v} - \uvecinprod{u}{w}\text{.}\)
-
\(\inprod{\uvec{u} + \uvec{v}}{\uvec{w}} = \uvecinprod{u}{w} + \uvecinprod{v}{w}\text{.}\)
-
\(\inprod{\uvec{u} - \uvec{v}}{\uvec{w}} = \uvecinprod{u}{w} - \uvecinprod{v}{w}\text{.}\)
-
\(\inprod{k \uvec{u}}{\uvec{v}} = k \uvecinprod{u}{v}\text{.}\)
-
\(\inprod{\uvec{u}}{k \uvec{v}} = k \uvecinprod{u}{v}\text{.}\)
-
\(\uvecinprod{v}{v} = \unorm{v}^2\text{.}\)
-
Both \(\inprod{\zerovec}{\uvec{v}} = 0\) and \(\inprod{\uvec{v}}{\zerovec} = 0\text{.}\)
Proof.
We will prove only Rule 9 and leave the rest up to you, the reader. (Though some of these are just restatements of the inner product axioms.)
Following the advice of Discovery 36.4, consider
\begin{equation*}
\inprod{\zerovec}{\uvec{v}} = \inprod{k \zerovec}{\uvec{v}} = k \inprod{\zerovec}{\uvec{v}} \text{,}
\end{equation*}
which holds for every scalar \(k\) by Rule 6 (or, really, by Axiom RIP 3). If we take scalar \(k = 0\text{,}\) then we get
\begin{equation*}
\inprod{\zerovec}{\uvec{v}} = 0 \inprod{\zerovec}{\uvec{v}} = 0 \text{,}
\end{equation*}
as desired.
The symmetric formula \(\inprod{\uvec{v}}{\zerovec} = 0\) then follows by Rule 1 (or, really, by Axiom RIP 1).
Most of the rules above also hold for complex inner products, but there are some wrinkles due to the complex conjugate.
Proposition 36.6.2. Algebra rules of complex inner products.
Suppose \(\inprod{\blank}{\blank}\) is a complex inner product. Then the following are true for all vectors \(\uvec{u}\text{,}\) \(\uvec{v}\text{,}\) and \(\uvec{w}\) in the associated inner product space, and all scalars \(a\text{.}\)
-
\(\uvecinprod{v}{u} = \lcconj{\uvecinprod{u}{v}}\text{.}\)
-
\(\inprod{\uvec{u}}{\uvec{v} + \uvec{w}} = \uvecinprod{u}{v} + \uvecinprod{u}{w}\text{.}\)
-
\(\inprod{\uvec{u}}{\uvec{v} - \uvec{w}} = \uvecinprod{u}{v} - \uvecinprod{u}{w}\text{.}\)
-
\(\inprod{\uvec{u} + \uvec{v}}{\uvec{w}} = \uvecinprod{u}{w} + \uvecinprod{v}{w}\text{.}\)
-
\(\inprod{\uvec{u} - \uvec{v}}{\uvec{w}} = \uvecinprod{u}{w} - \uvecinprod{v}{w}\text{.}\)
-
\(\inprod{a \uvec{u}}{\uvec{v}} = a \uvecinprod{u}{v}\text{.}\)
-
\(\inprod{\uvec{u}}{a \uvec{v}} = \cconj{a} \uvecinprod{u}{v}\text{.}\)
-
\(\uvecinprod{v}{v} = \unorm{v}^2\text{.}\)
-
Both \(\inprod{\zerovec}{\uvec{v}} = 0\) and \(\inprod{\uvec{v}}{\zerovec} = 0\text{.}\)
Proof.
We will prove only Rule 7 and leave the rest up to you, the reader. (Though some of these are just restatements of the inner product axioms.)
Calculate
\begin{align*}
\inprod{\uvec{u}}{a \uvec{v}}
\amp = \lcconj{\inprod{a \uvec{v}}{\uvec{u}}}
\amp \amp\text{(i)}\\
\amp = \lcconj{a \inprod{\uvec{v}}{\uvec{u}}} \amp \amp\text{(ii)} \\
\amp = \cconj{a} \lcconj{\inprod{\uvec{v}}{\uvec{u}}} \amp \amp\text{(iii)} \\
\amp = \cconj{a} \uvecinprod{u}{v} \amp \amp\text{(iv)} \text{,}
\end{align*}
with justifications
-
Rule 1 (or, really, Axiom CIP 1);
-
Rule 6 (or, really, Axiom CIP 3);
-
Rule 3 of Proposition A.2.11; and
-
Rule 1 (or, really, Axiom CIP 1).
The addition and scalar multiple rules for inner products can be combined to demonstrate that a real inner product is bilinear (i.e. linear in both terms), while a complex inner product is sesquilinear (i.e. linear in the first term but conjugate-linear in the second).
Aside: Terminology.
Proposition 36.6.3. Linearity of inner products.
-
A real inner product is bilinear. That is, for all vectors \(\uvec{u}\text{,}\) \(\uvec{v}\text{,}\) \(\uvec{w}\) in a real inner product space, and all scalars \(a\) and \(b\text{,}\) we have\begin{align*} \inprod{a \uvec{u} + b \uvec{v}}{\uvec{w}} \amp = a \uvecinprod{u}{w} + b \uvecinprod{v}{w},\\ \inprod{\uvec{u}}{a \uvec{v} + b \uvec{w}} \amp = a \uvecinprod{u}{v} + b \uvecinprod{u}{w}. \end{align*}
-
A complex inner product is sesquilinear. That is, for all vectors \(\uvec{u}\text{,}\) \(\uvec{v}\text{,}\) \(\uvec{w}\) in a complex inner product space, and all scalars \(a\) and \(b\text{,}\) we have\begin{align*} \inprod{a \uvec{u} + b \uvec{v}}{\uvec{w}} \amp = a \uvecinprod{u}{w} + b \uvecinprod{v}{w},\\ \inprod{\uvec{u}}{a \uvec{v} + b \uvec{w}} \amp = \cconj{a} \uvecinprod{u}{v} + \cconj{b} \uvecinprod{u}{w}. \end{align*}
The properties of norm that we encountered using the dot product in \(\R^n\) remain true in every inner product space.
Proposition 36.6.4. Properties of the norm.
The following are true for all vectors \(\uvec{u}\) and \(\uvec{v}\) in an inner product space (real or complex) and all scalars \(k\text{.}\)
-
\(\unorm{v} \ge 0\text{,}\) and \(\unorm{v} = 0\) only for \(\uvec{v} = 0\text{.}\)
-
\(\norm{-\uvec{v}} = \unorm{v}\text{.}\)
-
\(\norm{k\uvec{v}} = \abs{k}\unorm{v}\text{,}\) where \(\abs{k}\) indicates the ordinary absolute value of \(k\) in the context of a real inner product space, but indicates the complex modulus of \(k\) in the context of a complex inner product space.
-
\(\norm{\uvec{v}-\uvec{u}} = \norm{\uvec{u}-\uvec{v}}\text{.}\)
Both the Cauchy-Schwarz inequality (Theorem 13.5.4) and the Triangle inequality (Theorem 13.5.6) remain true in every inner product space, real or complex.
Theorem 36.6.5. Cauchy-Schwarz inequality.
For every pair of vectors \(\uvec{u}\) and \(\uvec{v}\) in an inner product space, we have \(\abs{\udotprod{u}{v}} \le \unorm{u}\unorm{v}\text{,}\) where \(\abs{\udotprod{u}{v}}\) indicates the ordinary absolute value in the real context and the complex modulus in the complex context.
Theorem 36.6.6. Triangle inequality.
For every pair of vectors \(\uvec{u}\) and \(\uvec{v}\) in an inner product space, we have \(\norm{\uvec{u}+\uvec{v}} \le \unorm{u} + \unorm{v}\text{.}\)
A vector can always be normalized to a unit vector.
Proposition 36.6.7. Normalization.
For nonzero vector \(\uvec{v}\) in an inner product space, the unit vectors in \(\Span \{\uvec{v}\}\) are
\begin{equation*}
\frac{k}{\unorm{v}} \uvec{v} \text{,}
\qquad \text{for} \qquad
\abs{k} = 1\text{.}
\end{equation*}
Proof idea.
Just calculate
\begin{equation*}
\norm{\frac{k}{\unorm{v}} \uvec{v}}
\end{equation*}
Remark 36.6.8.
As usual, in the above proposition \(\abs{k}\) indicates the ordinary absolute value in the real context and the complex modulus in the complex context. The only two real scalars that satisfy \(\abs{k} = 1\) are \(k = \pm 1\text{.}\) But in the complex context, there are an infinite number of scalars that satisfy this condition: the entire unit circle in the complex plane.
Subsection 36.6.2 Inner products of \(\R^n\) and \(\C^n\)
Finally we’ll address what we learned about alternative inner products in Discovery 36.7 and Subsection 36.4.7.
Proposition 36.6.9. Modified dot products are inner products.
-
If \(A\) is a positive definite real matrix, then the pairing\begin{equation*} \uvecinprod{u}{v} = \utrans{\uvec{v}} A \uvec{u} \end{equation*}defines an inner product on \(\R^n\text{.}\)
-
If \(A\) is a positive definite complex matrix, then the pairing\begin{equation*} \uvecinprod{u}{v} = \adjoint{\uvec{v}} A \uvec{u} \end{equation*}defines an inner product on \(\C^n\text{.}\)
Proof outline.
In the real case, assuming \(A\) is positive definite means, by definition, that \(A\) is symmetric and that the pairing defined in the statement satisfies Axiom RIP 4. The symmetry of \(A\) addresses Axiom RIP 1, which leaves only Axiom RIP 2 and Axiom RIP 3, which are easily confirmed using the properties of matrix algebra (Proposition 4.5.1).
The complex case is similar, with self-adjointness taking the place of symmetry.
The converse of the above proposition is true as well.
Theorem 36.6.10. Inner products are modified dot products.
-
If \(\inprod{\blank}{\blank}\) is an inner product on \(\R^n\text{,}\) then there exists a positive definite real matrix \(A\) such that\begin{equation*} \uvecinprod{u}{v} = \utrans{\uvec{v}} A \uvec{u} \end{equation*}for all \(\uvec{u},\uvec{v}\) in \(\R^n\text{.}\)
-
If \(\inprod{\blank}{\blank}\) is an inner product on \(\C^n\text{,}\) then there exists a positive definite complex matrix \(A\) such that\begin{equation*} \uvecinprod{w}{z} = \adjoint{\uvec{z}} A \uvec{w} \end{equation*}for all \(\uvec{w},\uvec{z}\) in \(\C^n\text{.}\)
Proof.
We will prove only the real case; the complex case is similar.
First, notice that for every matrix \(A\text{,}\)
\begin{equation*}
\utrans{\uvec{e}}_i A \uvec{e}_j = a_{ij} \text{,}
\end{equation*}
where \(a_{ij}\) is the \((i,j)\) entry of \(A\) and \(\uvec{e}_i,\uvec{e}_j\) are standard basis vectors, as usual. So if we set \(A\) to be the matrix with entries
\begin{equation*}
a_{ij} = \inprod{\uvec{e}_j}{\uvec{e}_i} \text{,}
\end{equation*}
then we have
\begin{equation*}
\inprod{\uvec{e}_i}{\uvec{e}_j} = a_{ji} = \utrans{\uvec{e}}_j A \uvec{e}_i \text{,}
\end{equation*}
which is a start. And, as usually is the case, what is true for basis vectors will be true for all vectors.
Aside: Notice.
Let’s check: given \(\uvec{u},\uvec{v}\) in \(\R^n\text{,}\) we have
\begin{align*}
\utrans{\uvec{v}} A \uvec{u}
\amp = \utrans{(v_1 \uvec{e}_1 + \dotsb + v_n \uvec{e}_n)} A (u_1 \uvec{e}_1 + \dotsb + u_n \uvec{e}_n)\\
\amp = (v_1 \utrans{\uvec{e}}_1 + \dotsb + v_n \utrans{\uvec{e}}_n) (u_1 A \uvec{e}_1 + \dotsb + u_n A \uvec{e}_n)\\
\amp = \sum_{i=1}^n \sum_{j=1}^n v_j u_i \utrans{\uvec{e}}_j A \uvec{e}_i\\
\amp = \sum_{i=1}^n \sum_{j=1}^n u_i v_j \inprod{\uvec{e}_i}{\uvec{e}_j}\\
\amp = \inprod{u_1 \uvec{e}_1 + \dotsb + u_n \uvec{e}_n}{v_1 \uvec{e}_1 + \dotsb + v_n \uvec{e}_n}\\
\amp = \uvecinprod{u}{v}\text{,}
\end{align*}
as desired.
It only remains to verify that \(A\) is positive definite. Using Axiom RIP 1, we have
\begin{equation*}
a_{ij} = \inprod{\uvec{e}_j}{\uvec{e}_i}
= \inprod{\uvec{e}_i}{\uvec{e}_j}
= a_{ji}\text{,}
\end{equation*}
so \(A\) is symmetric. And the fact that
\begin{equation*}
\utrans{\uvec{v}} A \uvec{u} = \uvecinprod{u}{v} \text{,}
\end{equation*}
that we have verified above, guarantees that
\begin{equation*}
\utrans{\uvec{v}} A \uvec{v} \gt 0
\end{equation*}
for all nonzero \(\uvec{v}\text{,}\) by Axiom RIP 4.
Finally, we verify the method of constructing positive definite matrices discussed in Subsection 36.4.7.
Proposition 36.6.11. Constructing positive definite matrices.
-
If \(P\) is an invertible real matrix, then \(A = \utrans{P} P\) is a positive definite real matrix.
-
If \(P\) is an invertible complex matrix, then \(A = \adjoint{P} P\) is positive definite complex matrix.
Proof.
Again, we will provide the proof only in the real context; the complex context is similar.
It is straightfoward to verify that \(A = \utrans{P} P\) is symmetric by showing \(\utrans{A} = A\text{.}\) So we will focus on the positive part of the definition of positive definite real matrix. Suppose \(\uvec{x}\) is a nonzero column vector in \(\R^n\text{.}\) Then,
\begin{align*}
\utrans{\uvec{x}} A \uvec{x} \amp = \utrans{\uvec{x}} \utrans{P} P \uvec{x} \\
\amp = \utrans{(P \uvec{x})} (P \uvec{x}) \\
\amp = \dotprod{(P \uvec{x})}{(P \uvec{x})} \\
\amp = \norm{P \uvec{x}}^2 \text{.}
\end{align*}
We assumed both \(P\) invertible and \(\uvec{x} \neq \zerovec\text{,}\) so \(P \uvec{x} \neq \zerovec\) must be true as well (Theorem 6.5.2). Which also means \(\norm{P \uvec{x}} \neq 0\) (Rule 1 of Proposition 36.6.4). And therefore
\begin{equation*}
\utrans{\uvec{x}} A \uvec{x} = \norm{P \uvec{x}}^2 > 0 \text{,}
\end{equation*}
as desired.
