(34S) The Complex Transformations Between ...

[Thomas Klemm]

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