(34S) The Complex Transformations Between ...
05-20-2014, 09:57 AM (This post was last modified: 06-15-2017 01:22 PM by Gene.)
Post: #1
 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
(34S) The Complex Transformations Between ...
For an introduction see: Calculations With Complex Matrices

These transformations convert between the $$Z^P$$ representation of an m×n complex matrix and a 2m×2n partitioned matrix of the following form:

$Z = \begin{bmatrix}X & -Y \\ Y & X\end{bmatrix}$

The matrix $$\tilde{Z}$$ created by the MPZ transformation has twice as many elements as $$Z^P$$.
For example, the matrices below show how $$\tilde{Z}$$ is related to $$Z^P$$.

$Z^P = \begin{bmatrix}1 & -6 \\ -4 & 5\end{bmatrix}$

$\tilde{Z} = \begin{bmatrix}1 & -6 & 4 & -5 \\ -4 & 5 & 1 & -6\end{bmatrix}$

The transformations that convert the representation of a complex matrix between $$Z^P$$ and $$\tilde{Z}$$ are shown in the following table.

$\begin{matrix} Pressing & Transforms & Into \\ MPZ & Z^P & \tilde{Z} \\ MZP & \tilde{Z} & Z^P \end{matrix}$

HP-15C
Owner's Handbook
Section 12: Calculating With Matrices
Calculating With Complex Matrices

Inverting a Complex Matrix

Example: (pp. 165)

$Z = \begin{bmatrix} 4+3i & 7-2i \\ 1+5i & 3+8i \end{bmatrix}$

Enter matrix $$Z^C$$:
7.0204
XEQ 'MED'
4 R/S
↓ 3 R/S
↓ 7 R/S
(…)
↓ 8 R/S

$Z^C = \begin{bmatrix}4 & 3 & 7 & -2 \\ 1 & 5 & 3 & 8\end{bmatrix}$

Transform matrix $$Z^C$$ to $$Z^P$$:
7.0204
XEQ'MCP'
7.0402

$Z^P = \begin{bmatrix}4 & 7 \\ 1 & 3 \\ 3 & -2 \\ 5 & 8\end{bmatrix}$

Transform matrix $$Z^P$$ to $$\tilde{Z}$$:
7.0402
XEQ'MPZ'
7.0404

$\tilde{Z}=\begin{bmatrix} 4 & 7 & -3 & 2 \\ 1 & 3 & -5 & -8 \\ 3 & -2 & 4 & 7 \\ 5 & 8 & 1 & 3 \end{bmatrix}$

Calculate the inverse:
7.0404
MATRIX:M-1
7.0404

$\bar{Z}=\tilde{Z}^{-1}=\begin{bmatrix} -0.0254 & 0.2420 & 0.2829 & 0.0022 \\ -0.0122 & -0.1017 & -0.1691 & 0.1315 \\ -0.2829 & -0.0022 & -0.0254 & 0.2420 \\ 0.1691 & -0.1315 & -0.0122 & -0.1017 \end{bmatrix}$

Transform matrix $$\bar{Z}$$ to $$\bar{Z}^P$$:
7.0404
XEQ'MZP'
7.0402

$\bar{Z}^P=\begin{bmatrix} -0.0254 & 0.2420 \\ -0.0122 & -0.1017 \\ -0.2829 & -0.0022 \\ 0.1691 & -0.1315 \end{bmatrix}$

Transform matrix $$\bar{Z}^P$$ to $$\bar{Z}^C$$:
7.0402
XEQ'MPC'
7.0204

$\bar{Z}^C=\begin{bmatrix} -0.0254 & -0.2829 & 0.2420 & -0.0022 \\ -0.0122 & 0.1691 & -0.1017 & -0.1315 \end{bmatrix}$

$Z^{-1}=\begin{bmatrix} -0.0254 - 0.2829i & 0.2420 - 0.0022i \\ -0.0122 + 0.1691i & -0.1017 - 0.1315i \end{bmatrix}$

Code:
LBL'MPZ' STO A STO B FP SDL 002 FP STO C RCL L IP # 002 / SDR 002 STO- B × SDL 004 STO D RCL+ B ENTER^ RCL+ D M.COPY # 002 +/- x<> Y ENTER^ M+× RCL+ D RCL B x<> Y M.COPY RCL A RCL+ D RCL+ D TRANSP RCL A TRANSP RCL+ C TRANSP END

Code:
LBL'MZP' TRANSP ENTER^ FP SDL 002 IP # 002 / SDR 002 - TRANSP END

Code:
LBL'MPC' IP STO A RCL L FP SDL 002 FP STO B STO+ B RCL L IP # 002 / STO+ B × # 002 + SDR 002 RCL+ A TRANSP RCL B SDR 002 RCL+ A END

Code:
LBL'MCP' IP STO A RCL L FP SDL 002 IP STO B STO+ B RCL L FP # 002 / STO+ B × # 002 SDR 004 + RCL+ A TRANSP RCL B SDR 002 RCL+ A END

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