为何MATLAB复向量点积定义与通用定义为复共轭关系？

Great question! This difference isn't a bug or oversight—it all comes down to conventions rooted in MATLAB's engineering-focused origins versus the pure math framing you'll often see on Wikipedia. Let's break this down clearly:

First, clarify the definition mismatch #

• The general complex dot product you'll find on Wikipedia (often from pure linear algebra contexts) is typically:

  $\text{dot}(A, B) = \sum_{i} A_i \cdot \overline{B_i}$
  (taking the complex conjugate of the second vector)

• MATLAB's implementation via A'*B or dot(A, B) computes:

  $\text{dot}(A, B) = \sum_{i} \overline{A_i} \cdot B_i$
  (taking the complex conjugate of the first vector)
  These two results are direct complex conjugates of each other.

The core reason: Engineering priorities drive MATLAB's choice #

MATLAB was built for engineers, not just mathematicians, and fields like signal processing, control systems, and communications have long favored the convention MATLAB uses because it simplifies daily, high-frequency tasks:

• Instant vector energy calculation: The most common use of dot products in engineering is finding the energy of a signal (or the squared 2-norm of a vector). With MATLAB's convention, A'*A instantly gives you the sum of the squared magnitudes of A's elements—no extra conj() calls required. This is critical for tasks like signal power analysis, error minimization, and system validation.
• Consistent matrix operations: MATLAB's default transpose operator ' is the Hermitian (conjugate) transpose (plain transpose uses .'). So A'*B is just a natural extension of standard matrix multiplication for complex data, which aligns with what engineers expect. No need to memorize separate rules for dot products versus matrix multiplies.

Why Wikipedia uses the alternate convention #

Pure math contexts (like abstract inner space theory) often define inner products with the conjugate applied to the second argument, but this is just a different starting point—both conventions satisfy the required properties of a complex inner product (e.g., $\langle u, v \rangle = \overline{\langle v, u \rangle}$). Wikipedia leans into the math-focused convention, but it's not "more correct"—just tailored to a different audience.

MATLAB's choice is practical for its core users #

By aligning the dot product with conjugate transpose-based matrix multiplication, MATLAB eliminates cognitive friction for its engineering user base. You don't have to switch between two distinct logics when moving from matrix operations to dot products, which saves time and reduces errors in day-to-day work.

内容的提问来源于stack exchange，提问作者java4ever