0
$\begingroup$

Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude.

I have noticed that $$\operatorname{tr}\bigg\{\left(\mathbf{X^{-1}\cdot p\cdot p^H\cdot \left(X^H\right)^{-1}}\right)\bigg\}\tag{1}$$ is related into the amplitude of $\mathbf p$. where $\operatorname{tr}$ is the trace operator.

  • For example if $\mathbf p = 1$; $$\implies\operatorname{tr}\bigg\{\left(\mathbf{X^{-1}\cdot p\cdot p^H\cdot \left(X^H\right)^{-1}}\right)\bigg\} = 2N$$

  • On the other hand, if $\mathbf p=0.5$; $$\implies\operatorname{tr}\bigg\{\left(\mathbf{X^{-1}\cdot p\cdot p^H\cdot \left(X^H\right)^{-1}}\right)\bigg\} = \frac N2$$

  • And so on.

My question, is there a general formula expressing the relationship between the equation $(1)$ and the amplitude of $\mathbf p$ ?

$\endgroup$

1 Answer 1

0
$\begingroup$

The trace of a matrix just scales with the scaling of the matrix:

$$\textrm{tr}\{c\mathbf{A}\}=c\,\textrm{tr}\{\mathbf{A}\}\tag{1}$$

So if you scale the vector $\mathbf{p}$, the trace scales with the squared magnitude of the scaling constant. With $\mathbf{p}=c\mathbf{p}_0$ you have

$$\begin{align}\textrm{tr}\{\mathbf{Ap}\mathbf{p}^H\mathbf{A}^H\}&=\textrm{tr}\{\mathbf{A}c\mathbf{p}_0c^*\mathbf{p}_0^H\mathbf{A}^H\}\\&=\textrm{tr}\{|c|^2\mathbf{A}\mathbf{p}_0\mathbf{p}_0^H\mathbf{A}^H\}\\&=|c|^2\textrm{tr}\{\mathbf{A}\mathbf{p}_0\mathbf{p}_0^H\mathbf{A}^H\}\end{align}$$

$\endgroup$
6
  • $\begingroup$ Thanks for your reply, .. but do you mean that $tr{AP_0P_0^HA^H}$ will be constant ? and finally, is it $(A^H)^{-1}$ = $(A^{-1})^H$ ? $\endgroup$ Nov 4, 2020 at 10:25
  • $\begingroup$ @New_student: $c$ is just a constant with which you scale the vector. So have a fixed vector $p_0$ and the final vector you use is $p=cp_0$. $A$ is just any matrix, it is irrelevant. In your case $A=X^{-1}$. And yes to your last question. $\endgroup$
    – Matt L.
    Nov 4, 2020 at 10:29
  • $\begingroup$ Thanks again, In that case $tr(AP_0P_0^HA^H)$ if $A$ is diagonal with diagonal elements $0.5+0.5i$ and $P_0 = 1$ should be constant, right? $\endgroup$ Nov 4, 2020 at 10:34
  • $\begingroup$ Could you please recheck the last equation? because when I made for example $tr(AP_0P_0^HA^H) = 8$ when $P_0 = 1$. Then I changed $P_0 = 0.5$ which means according to your last equality that $|0.5|^2tr(AP_0P_0^HA^H) = 2$ and that's right, but when made $P_0 = 0.25$, $|0.25|^2tr(AP_0P_0^HA^H) = 0.5$ however when doing it in MATLAB, it's $0.125$. I think something is missing in last equation. $\endgroup$ Nov 4, 2020 at 10:43
  • $\begingroup$ @New_student: The equation looks correct. $\endgroup$
    – Matt L.
    Nov 4, 2020 at 11:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.