Conventions For Complex Inner Products

Why Complex Inner-Product Conventions Need Care

Over \(\mathbb{R}\), switching the order of the arguments of an inner product does not change its value. Over \(\mathbb{C}\), switching the arguments conjugates the value. As a result, the two arguments do not obey the same scalar rule: if an inner product is linear in one argument, it is conjugate-linear in the other. Textbooks differ on which argument is chosen to be linear, so a formula must be read together with its convention.

Throughout this tutorial, \(V\) is a vector space over \(\mathbb{C}\). A complex number \(a\) has conjugate \(\overline{a}\). A map \(\langle\cdot,\cdot\rangle:V\times V\to\mathbb{C}\) is an inner product under the first-argument-linear convention if it is linear in its first argument, satisfies conjugate symmetry, and is positive definite. Here linearity means additivity and homogeneity with respect to complex scalars.

The second-argument rule is not an additional independent choice. It follows from the first-argument rule and conjugate symmetry. Likewise, the convention in which the second argument is linear follows by reversing the arguments. Deriving these rules is useful because it prevents accidental use of real-number algebra with complex scalars.

Deriving the Conjugate-Linear Rule

Proof. Conjugate symmetry lets us reverse the arguments, apply first-argument linearity, and reverse them back. In full, for \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\) and \(a,b\in\mathbb{C}\),

In the second line, first-argument linearity applies. In the third, complex conjugation conjugates each scalar and distributes over sums. The last line is conjugate symmetry applied to each pairing. This proves additivity and conjugate homogeneity in the second argument. \(\square\)

Thus, when the first argument is linear, multiplying the second argument by \(a\) multiplies the inner product by \(\overline{a}\), not by \(a\). If \(a\) is real, then \(\overline{a}=a\), which is one reason this distinction can go unnoticed when working only over \(\mathbb{R}\).

The Other Standard Convention

Another common choice is to make the second argument linear. Denote its pairing temporarily by \([\mathbf{u},\mathbf{v}]\), to distinguish it from the first-argument-linear pairing \(\langle\mathbf{u},\mathbf{v}\rangle\). The second convention has its own conjugate symmetry and positivity conditions. Its first-argument rule is then conjugate-linear.

The two conventions describe the same underlying pairing rule with opposite argument order. More precisely, given a first-argument-linear inner product, define a second-argument-linear one by \([\mathbf{u},\mathbf{v}]=\langle\mathbf{v},\mathbf{u}\rangle\). Conjugate symmetry also gives \([\mathbf{u},\mathbf{v}]=\overline{\langle\mathbf{u},\mathbf{v}\rangle}\). The following result verifies that this translation satisfies all the required rules.

Proof. For \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\) and \(a,b\in\mathbb{C}\), first-argument linearity of the original pairing gives \([\mathbf{u},a\mathbf{v}+b\mathbf{w}]=\langle a\mathbf{v}+b\mathbf{w},\mathbf{u}\rangle=a\langle\mathbf{v},\mathbf{u}\rangle+b\langle\mathbf{w},\mathbf{u}\rangle=a[\mathbf{u},\mathbf{v}]+b[\mathbf{u},\mathbf{w}]\). For conjugate symmetry, \([\mathbf{v},\mathbf{u}]=\langle\mathbf{u},\mathbf{v}\rangle=\overline{\langle\mathbf{v},\mathbf{u}\rangle}=\overline{[\mathbf{u},\mathbf{v}]}\). Finally, \([\mathbf{u},\mathbf{u}]=\langle\mathbf{u},\mathbf{u}\rangle\), so it is real and nonnegative and is positive whenever \(\mathbf{u}\ne\mathbf{0}\). These are exactly the second-argument convention’s requirements. \(\square\)

Expanding Pairings Without Losing Conjugates

Once the scalar rules are fixed, expansions follow systematically. Under the first-argument-linear convention, scalar coefficients in the first argument stay unchanged, while coefficients in the second argument are conjugated. Under the second-argument-linear convention, the roles reverse. Keeping a short formula nearby is safer than trying to infer the rules from real-valued examples.

Why Linearity in Both Arguments Fails

It may seem that both arguments could simply be made linear. Conjugate symmetry and positive definiteness rule this out on every nonzero complex inner-product space. The qualification “nonzero” matters: the zero vector space has no nonzero vectors on which the contradiction could be tested.

Proof. First suppose \(V\) is nonzero and choose \(\mathbf{x}\ne\mathbf{0}\). Positive definiteness gives \(r=\langle\mathbf{x},\mathbf{x}\rangle>0\), with \(r\) real. If the pairing were linear in both arguments, then \(\langle i\mathbf{x},\mathbf{x}\rangle=ir\) by first-argument linearity. Conjugate symmetry would then imply \(\langle\mathbf{x},i\mathbf{x}\rangle=\overline{\langle i\mathbf{x},\mathbf{x}\rangle}=\overline{ir}=-ir\). But second-argument linearity also gives \(\langle\mathbf{x},i\mathbf{x}\rangle=i\langle\mathbf{x},\mathbf{x}\rangle=ir\). Hence \(ir=-ir\), so \(2ir=0\), contradicting \(r>0\). For the zero vector space, the only possible pairing has value \(0\) on its sole pair \((\mathbf{0},\mathbf{0})\). Both linearity rules and conjugate symmetry hold, and positive definiteness requires positivity only for nonzero vectors, of which there are none. \(\square\)

Common Mistake: Treating Conjugate Symmetry as Symmetry

A frequent error is to write \(\langle\mathbf{u},\mathbf{v}\rangle=\langle\mathbf{v},\mathbf{u}\rangle\) over \(\mathbb{C}\), copying the real inner-product rule. For the first-argument-linear pairing above, \(z=2+i\) and \(w=1-2i\) give \(\langle z,w\rangle=5i\), while \(\langle w,z\rangle=-5i\). The incorrect symmetry claim would assert \(5i=-5i\), which is false. The correction is conjugate symmetry: \(\langle z,w\rangle=\overline{\langle w,z\rangle}=\overline{-5i}=5i\).

Another version of the same error is to use the first-argument scalar rule in the second argument. For example, \(\langle z,aw\rangle\) is not generally \(a\langle z,w\rangle\) under the first-argument-linear convention; it is \(\overline{a}\langle z,w\rangle\). This distinction disappears for real scalars but matters for complex ones.

When translating a formula, first identify the convention, then apply the scalar rule for each argument separately. Reversing the arguments converts one convention to the other, and conjugate symmetry converts the value. These checks are especially important in computations involving complex scalars, where swapping arguments or moving a scalar without conjugating it can change the result.