derivative of h^{*} w.r.t h where h is a 1 by N complex vector

To compute the derivative of \( h^{*} \) with respect to \( h \), where \( h \) is a \( 1 \times N \) complex vector, we can proceed step by step. Here, \( h^{*} \) represents the complex conjugate transpose (or Hermitian transpose) of the vector \( h \).

### Step-by-step explanation:

1. Let \( h \) be a \( 1 \times N \) complex vector:
   \[
   h = \begin{bmatrix} h_1 & h_2 & \cdots & h_N \end{bmatrix}
   \]
   where each \( h_i \) is a complex number.

2. The Hermitian (conjugate transpose) of \( h \), denoted \( h^{*} \), is an \( N \times 1 \) column vector:
   \[
   h^{*} = \begin{bmatrix} \overline{h_1} \\ \overline{h_2} \\ \vdots \\ \overline{h_N} \end{bmatrix}
   \]
   where \( \overline{h_i} \) is the complex conjugate of \( h_i \).

3. Now, we need to compute \( \frac{\partial h^{*}}{\partial h} \).

### General rule for complex derivatives:
When differentiating a complex function, we need to treat the complex variable \( h \) and its conjugate \( \overline{h} \) as independent variables.

The derivative of a complex conjugate \( \overline{h_i} \) with respect to \( h_j \) is given by:
\[
\frac{\partial \overline{h_i}}{\partial h_j} = 0 \quad \text{if} \ i \neq j
\]
\[
\frac{\partial \overline{h_i}}{\partial h_i} = 0
\]

Therefore, the derivative of \( h^{*} \) with respect to \( h \) (as a vector) is a matrix of zeros. This can be represented as:
\[
\frac{\partial h^{*}}{\partial h} = 0
\]

### Conclusion:
The derivative of \( h^{*} \) with respect to \( h \), where \( h \) is a \( 1 \times N \) complex vector, is a zero matrix.

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Would you like more details on this? Here are some related questions that might help:

1. What are the rules for complex matrix calculus?
2. How does one differentiate complex functions involving both \( h \) and \( h^* \)?
3. Can you explain the Wirtinger derivatives in complex analysis?
4. How would the result change if \( h \) was real instead of complex?
5. How do complex derivatives apply in machine learning or signal processing?

**Tip:** When differentiating complex vectors or matrices, always treat the variables and their conjugates as independent unless explicitly stated otherwise.

This article provides a step-by-step solution to compute the derivative of the conjugate transpose (h*) with respect to a 1xN complex vector h. Explore concepts like complex vectors, matrix calculus, and key rules for complex derivatives. Suitable for undergraduate mathematics students.

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