Recipe

Compute an inner product and confirm positive definiteness

Use when: you're given vectors in $\mathbb R^n$ or $\mathbb C^n$ and need $\langle\mathbf u,\mathbf v\rangle$ computed correctly, or need to confirm a pairing is genuinely positive definite

Goal: the correct numeric value of $\langle\mathbf u,\mathbf v\rangle$ , together with confidence that pairing a vector with itself is real and positive

Taught in Inner products over R and C

Quick recipe

Decide whether the space is real or complex
Multiply term by term and add
When checking positivity, evaluate at $\mathbf u=\mathbf v$ and confirm the result

1
Check whether the vectors' entries are real numbers or genuinely involve $i$ ; this decides which formula applies.
- the space is real — use $\langle\mathbf u,\mathbf v\rangle=\sum_i u_iv_i$ , the plain dot product, with no conjugation needed.
- the space is complex — use $\langle\mathbf u,\mathbf v\rangle=\sum_i u_i\overline{v_i}$ , conjugating every entry of the SECOND vector first.
Why: the complex formula's conjugation is what keeps $\langle\mathbf v,\mathbf v\rangle$ real and nonnegative, and skipping it over $\mathbb C$ breaks positive definiteness entirely.
2
Multiply each entry of $\mathbf u$ against the matching (possibly conjugated) entry of $\mathbf v$ , then sum the results.

Example

For $\mathbf u=(1,2i)$ , $\mathbf v=(i,1)$ : $\langle\mathbf u,\mathbf v\rangle=(1)\overline{i}+(2i)\overline{1}=(1)(-i)+(2i)(1)=-i+2i=i$ .

Why: this term-by-term sum is exactly what the formula from Step 1 means, carried out on actual numbers instead of left symbolic.
3
Substitute the same vector into both slots using the correct formula from Step 1, then check the result is real and strictly positive for a nonzero vector.

Example

For $\mathbf v=(3,-4)$ in $\mathbb R^2$ : $\langle\mathbf v,\mathbf v\rangle=3^2+(-4)^2=9+16=25>0$ .

For $\mathbf u=(1,i)$ using the WRONG plain formula $\sum u_i^2$ : $1^2+i^2=1+(-1)=0$ , despite $\mathbf u\ne\mathbf 0$ .
- the correctly conjugated formula was used — the result is guaranteed real and nonnegative, strictly positive whenever $\mathbf v\ne\mathbf 0$ .
- the plain, unconjugated formula was used on a complex vector by mistake — the result can come out zero or non-real even when $\mathbf v\ne\mathbf 0$ .
Why: conjugating one slot is exactly what forces $\langle\mathbf v,\mathbf v\rangle$ onto the nonnegative real axis, so dropping it is the one mistake that breaks positive definiteness.

Common mistakes

Using the plain dot product formula on complex vectors, forgetting to conjugate the second argument.
Conjugating the FIRST argument instead of the second, which flips the sign of the imaginary part in the final answer.
Assuming any one positive real result confirms positive definiteness, rather than checking it holds for every nonzero vector.
Forgetting that $\langle\mathbf u,\mathbf v\rangle$ and $\langle\mathbf v,\mathbf u\rangle$ are complex conjugates of each other over $\mathbb C$ , not equal.