Inner Products: Definition and Properties

1. From Dot Product to Real Inner Product

You're already familiar with the dot product for vectors in ${\mathbb{R}}^n$. For two vectors $\mathbf{u}=(u_1,...,u_n)$ and $\mathbf{v}=(v_1,...,v_n)$, their dot product is $\mathbf{u}\cdot \mathbf{v}=u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n$.

Beyond just a computational formula, the dot product provides geometry. It allows us to define:

• Length (or Norm): The length of a vector $\mathbf{v}$ is $\Vert \mathbf{v}\Vert =\sqrt{\mathbf{v}\cdot \mathbf{v}}$.
• Angle: The angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by $\cos \theta =\frac{\mathbf{u}\cdot \mathbf{v}}{\Vert \mathbf{u}\Vert \Vert \mathbf{v}\Vert }$.
• Orthogonality: Two non-zero vectors are perpendicular if their dot product is zero.

The video below gives a great recap of why these geometric properties are so useful, setting the stage for why we want to abstract them.

Defining the Real Inner Product

An inner product on a real vector space $V$ is a function that takes two vectors $\mathbf{u},\mathbf{v}\in V$ and returns a real number, denoted $⟨\mathbf{u},\mathbf{v}⟩$, that satisfies the following three axioms:

1. Linearity in each argument (this property is called bilinearity):

   • $⟨c\mathbf{u}+d\mathbf{w},\mathbf{v}⟩=c⟨\mathbf{u},\mathbf{v}⟩+d⟨\mathbf{w},\mathbf{v}⟩$ for any scalars $c,d$.
   • $⟨\mathbf{u},c\mathbf{v}+d\mathbf{w}⟩=c⟨\mathbf{u},\mathbf{v}⟩+d⟨\mathbf{u},\mathbf{w}⟩$.

2. Symmetry:

   • $⟨\mathbf{u},\mathbf{v}⟩=⟨\mathbf{v},\mathbf{u}⟩$.

3. Positive-Definiteness:

   • $⟨\mathbf{v},\mathbf{v}⟩\ge 0$ for all vectors $\mathbf{v}$.
   • $⟨\mathbf{v},\mathbf{v}⟩=0$ if and only if $\mathbf{v}$ is the zero vector, $\mathbf{v}=0$.

A real vector space equipped with an inner product is called a real inner product space.

The following video walks through these formal definitions and shows how the standard dot product in ${\mathbb{R}}^n$ is a perfect example of a function that satisfies all these rules.

An Example Beyond ${\mathbb{R}}^n$: Inner Products on Function Spaces

Inner products are not limited to column vectors. We can define them for more abstract vector spaces, like the space of continuous functions you've worked with in calculus.

Consider the vector space $C[a,b]$ of all continuous real-valued functions on the interval $[a,b]$. We can define an inner product for two functions $f(x)$ and $g(x)$ as:

This might seem unusual, but this definition fulfills all the inner product axioms and is fundamental in fields like Fourier analysis and quantum mechanics (where wavefunctions are treated as vectors in a function space).

The Oxford Linear Algebra video you just watched continues with an example of this type.

2. The Complex (Hermitian) Inner Product

For quantum mechanics, we must work with vector spaces where the scalars are complex numbers. This requires a small but crucial modification to the inner product definition.

Why the Real Definition Fails

Let's see what goes wrong if we use the real inner product axioms for a complex vector space. The positive-definiteness axiom requires $⟨\mathbf{v},\mathbf{v}⟩\ge 0$, ensuring the "length squared" of any vector is a non-negative real number.

Consider a vector $\mathbf{v}$ in ${\mathbb{C}}^1$ (i.e., just a complex number). Let $\mathbf{v}=i$. If we tried to define an inner product as $⟨\mathbf{u},\mathbf{v}⟩=uv$ (analogous to the dot product), we would get:

A length-squared of -1 is problematic! This violates positive-definiteness. The fix involves complex conjugation.

Defining the Complex Inner Product

As the video showed, the solution is to replace the symmetry axiom. An inner product on a complex vector space $V$ (also called a Hermitian inner product) is a function $⟨\cdot ,\cdot ⟩$ that maps two vectors to a complex number and satisfies:

1. Linearity in the second argument:

   • $⟨\mathbf{u},c\mathbf{v}+d\mathbf{w}⟩=c⟨\mathbf{u},\mathbf{v}⟩+d⟨\mathbf{u},\mathbf{w}⟩$.

2. Conjugate Symmetry:

   • $⟨\mathbf{u},\mathbf{v}⟩=\overline{⟨\mathbf{v},\mathbf{u}⟩}$. (The bar denotes complex conjugation).

3. Positive-Definiteness:

   • $⟨\mathbf{v},\mathbf{v}⟩\ge 0$ and is a real number.
   • $⟨\mathbf{v},\mathbf{v}⟩=0$ if and only if $\mathbf{v}=0$.

A complex vector space with such an inner product is a complex inner product space (or a unitary space).

A Note on Conventions: Sesquilinearity

What about linearity in the first argument? Let's derive it from the axioms:

So, $⟨c\mathbf{u},\mathbf{v}⟩=\bar{c}⟨\mathbf{u},\mathbf{v}⟩$. Pulling a scalar out of the first argument requires conjugating it. This property is called conjugate linearity or anti-linearity. An operator that is linear in one argument and anti-linear in the other is called sesquilinear (from Latin for "one and a half times linear").

Important Convention Note: Physicists (especially in quantum mechanics) almost universally define the inner product to be linear in the second argument and anti-linear in the first, as we have done. Many mathematicians, however, use the opposite convention. Both are valid, but you must be consistent. We will stick to the physics convention as it aligns with your goal of studying quantum mechanics.

Standard Inner Product on ${\mathbb{C}}^n$

The standard inner product on ${\mathbb{C}}^n$ for vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as:

Using matrix notation, if $\mathbf{u}$ and $\mathbf{v}$ are column vectors, this is written as ${\mathbf{u}}^{†}\mathbf{v}$, where ${\mathbf{u}}^{†}$ (read "u-dagger") is the conjugate transpose (or Hermitian conjugate) of $\mathbf{u}$—you transpose the vector and take the complex conjugate of each element.

For $\mathbf{u}=\left(\begin{array}{c}{u}_1\\ ⋮\\ u_n\end{array}\right)$, ${\mathbf{u}}^{†}=({\bar{u}}_1,\dots ,{\bar{u}}_n)$.

Let's check the positive-definiteness for this definition:

Since $|v_i{|}^2$ is the squared modulus of a complex number, it is always a non-negative real number. The sum is therefore also a non-negative real number, and it can only be zero if all $|v_i{|}^2$ are zero, which means $\mathbf{v}=0$. This is exactly what we need!

The following resources provide a clear definition and verification of the complex inner product.

Conclusion

In this lesson, we have generalized the dot product to the more abstract and powerful concept of an inner product. This allows us to import geometric intuition into all kinds of vector spaces, which will be immensely valuable.

Key Takeaways:

• An inner product is a function that takes two vectors and produces a scalar, satisfying axioms of linearity, symmetry (for real spaces) or conjugate symmetry (for complex spaces), and positive-definiteness.
• The real inner product uses symmetry: $⟨\mathbf{u},\mathbf{v}⟩=⟨\mathbf{v},\mathbf{u}⟩$. The standard dot product is an example.
• The complex (Hermitian) inner product uses conjugate symmetry: $⟨\mathbf{u},\mathbf{v}⟩=\overline{⟨\mathbf{v},\mathbf{u}⟩}$. This is necessary to ensure that the inner product of a vector with itself, $⟨\mathbf{v},\mathbf{v}⟩$, is a non-negative real number.
• The standard inner product on ${\mathbb{C}}^n$ is $⟨\mathbf{u},\mathbf{v}⟩={\mathbf{u}}^{†}\mathbf{v}={\sum }_i{\bar{u}}_i{v}_i$.
• A vector space equipped with an inner product is called an inner product space.

Preview of the Next Lesson:

Now that we have formally defined the inner product, we can immediately put it to use. In the next lesson, we will use the positive-definiteness property $⟨\mathbf{v},\mathbf{v}⟩\ge 0$ to formally define the norm (or length) of a vector as $\Vert \mathbf{v}\Vert =\sqrt{⟨\mathbf{v},\mathbf{v}⟩}$. From there, we will define distances and explore their properties.