Dot product vs. inner product vs. bilinear form

Linear Algebra · Hierarchy

Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.

dot product ⊂ inner products ⊂ bilinear forms

Definitions

dot product

High school

Consider two vectors $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ in $\reals^n$. The dot product of $x$ and $y$ is

\[x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.\]

Undergraduate

Consider the vector space $\reals^n$, with vectors written as columns. The dot product is the standard inner product on $\reals^n$,

\[x \cdot y = x^{\top} y.\]

Examples

• $(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto x_1 y_1 + x_2 y_2$ is the dot product on $\reals^2$.

Non-examples

• $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is not the dot product, because of the factor $2$.

Properties

• $x \cdot y = \norm{x} \norm{y} \cos \theta$, where $\theta$ is the angle between $x$ and $y$. In particular, $x \cdot y = 0$ exactly when $x$ and $y$ are perpendicular (or one of them is zero).

inner product

High school

An inner product is a way of multiplying two vectors to get a number, generalizing the dot product, that gives a way to measure lengths and angles.

Undergraduate (real vector spaces)

Consider a real vector space $V$ and a function $\ip{\cdot}{\cdot} : V \times V \to \reals$. The function is an inner product if

1. it is linear in its first argument: $\ip{a x + y}{z} = a \ip{x}{z} + \ip{y}{z}$ for all $x, y, z \in V$ and $a \in \reals$,
2. it is symmetric: $\ip{x}{y} = \ip{y}{x}$ for all $x, y \in V$, and
3. it is positive definite: $\ip{x}{x} > 0$ for for all $x \neq 0 in $V$.

Caveat. For complex vector spaces, symmetry is replaced by conjugate symmetry, $\ip{x}{y} = \overline{\ip{y}{x}}$, so the inner product is linear in one argument and conjugate-linear in the other.

Graduate (real or complex vector spaces)

Consider a vector space $V$ over $\mathbb{F}$, where $\mathbb{F}$ is $\reals$ or $\mathbb{C}$, and a sesquilinear form $\ip{\cdot}{\cdot}$ on $V$: a function $V \times V \to \mathbb{F}$ that is linear in its first argument and conjugate-linear in its second. The form is an inner product if, for all $x, y \in V$,

1. it is Hermitian: $\ip{x}{y} = \overline{\ip{y}{x}}$, and
2. it is positive definite: $\ip{x}{x} > 0$ for $x \neq 0$.

Over $\reals$, a sesquilinear form is a bilinear form, and Hermitian means symmetric.

Examples

• $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is an inner product, because it is symmetric and positive definite.
• $(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto x_1 y_1 + x_2 y_2$ is an inner product: the dot product on $\reals^2$.

Non-examples

• $x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$
  $x \mapsto \abs{x_1} + \abs{x_2}$ does not come from an inner product, because it fails the parallelogram law: with $x = (1, 0)$ and $y = (0, 1)$, $\norm{x + y}^2 + \norm{x - y}^2 = 8$, but $2 \norm{x}^2 + 2 \norm{y}^2 = 4$.
• $(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto x_1 y_1 - x_2 y_2$ is not an inner product, because it is not positive definite: $(0, 1)$ paired with itself gives $-1$.

bilinear form

Undergraduate

Consider a vector space $V$. The function $B : V \times V \to \reals$ is a bilinear form on $V$ if $B$ is linear in each argument when the other is held fixed. It need not be symmetric or positive.

Examples

• $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is a bilinear form, because it is linear in $x$ and in $y$.
• $(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto x_1 y_1 - x_2 y_2$ is a bilinear form.
• $(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
  $(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$ is an inner product and a bilinear form, because it is the dot product.

Properties

• Every bilinear form on $\reals^n$ has the form $B(x, y) = x^{\top} A y$ for a unique matrix $A$, and $B$ is symmetric if and only if $A$ is.

Examples

Example	dot product	inner product	bilinear form	Why
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$	✓	✓	✓
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$	✗	✓	✓
$(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$	–	✗	✓

Notes

• A bilinear form $B$ is symmetric if $B(x, y) = B(y, x)$, and positive definite if $B(x, x) > 0$ for every $x \neq 0$.

• On a complex vector space, an inner product is conjugate symmetric and linear in only one argument (sesquilinear), so it is not a bilinear form.

See also

• Norm vs. metric vs. inner product

All distinctions · All definitions