05-18-2014, 05:20 PM
Introduction
This article describes how to perform matrix multiplication and matrix inversion with complex matrices. Since the WP-34S lacks complex numbers the same trick as in the HP-15C is used. The complex transformation between \(Z^P\) and \(\tilde{Z}\) is provided by combining M.COPY and M.TRANSP commands. Of course this could be written down in a small program.
MED: Matrix Editor
This description is from matrixedit.wp34s:
HP-15C
Owner's Handbook
Section 12: Calculating With Matrices
Calculating With Complex Matrices
Inverting a Complex Matrix
Example: (pp. 165)
\[Z = \begin{bmatrix} 4+3i & 7-2i \\ 1+5i & 3+8i \end{bmatrix}\]
\[Z^P = \begin{bmatrix}4 & 7 \\ 1 & 3 \\ 3 & -2 \\ 5 & 8\end{bmatrix}\]
Enter matrix \(Z^P\):
0.0402
XEQ 'MED'
4 R/S
↓ 7 R/S
↓ 1 R/S
(…)
↓ 8 R/S
Calculate offset for element (1,3):
1
3
0.0802
MATRIX:M.REG
4
Calculate offset for element (1,5):
1
5
0.0802
MATRIX:M.REG
8
Copy complex 2×2 matrix:
4.0202
8
MATRIX:M.COPY
8.0202
Negate 2×2 matrix (-x = x - 2x):
-2
4.0202
8.0202
MATRIX:M+x
Calculate offset for element (1,1):
1
1
0.0402
MATRIX:M.REG
0
Calculate offset for element (1,7):
1
7
0.0802
MATRIX:M.REG
12
Copy real 2×2 matrix:
0.0202
12
MATRIX:M.COPY
12.0202
Transpose upper 4×2 matrix:
0.0402
MATRIX:TRANSP
0.0204
Transpose lower 4×2 matrix:
8.0402
MATRIX:TRANSP
8.0202
Transpose 4×4 matrix:
0.0404
MATRIX:TRANSP
0.0404
\[\tilde{Z}=\begin{bmatrix} 4 & 7 & -3 & 2 \\ 1 & 3 & -5 & -8 \\ 3 & -2 & 4 & 7 \\ 5 & 8 & 1 & 3 \end{bmatrix}\]
Calculate the inverse:
0.0404
MATRIX:M-1
\[\tilde{Z}^{-1}=\begin{bmatrix} -0.0254 & 0.2420 & 0.2829 & 0.0022 \\ -0.0122 & -0.1017 & -0.1691 & 0.1315 \\ -0.2829 & -0.0022 & -0.0254 & 0.2420 \\ 0.1691 & -0.1315 & -0.0122 & -0.1017 \end{bmatrix}\]
Solving the Complex Equation AX = B
Example: (pp. 169)
\[A^P=\begin{bmatrix} 10 & 0 \\ 0 & 0 \\ 200 & -200 \\ -200 & 170 \end{bmatrix}\]
\[B^P=\begin{bmatrix} 5 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
Calculate inverse of AP as above and multiply by BP:
0.0404
16.0401
20
MATRIX:M×
20.0401
\[\tilde{A}^{-1}\times B^P=\begin{bmatrix} 0.0372 \\ 0.0437 \\ 0.1311 \\ 0.1543 \end{bmatrix}\]
Transpose 2×2 matrix:
20.0202
MATRIX:TRANSP
\[\begin{bmatrix} 0.0372 & 0.1311 \\ 0.0437 & 0.1543 \end{bmatrix}\]
HP-42S
RPN Scientific
Programming Examples and Techniques
5: Matrices
Solving Simultaneous Equations That Have Complex Terms
Example: (pp. 166)
\[A=\begin{bmatrix} 10-i0.01 & -5 & 0 & 0 \\ -5 & 15-i0.01 & -5 & 0 \\ 0 & -5 & 15-i0.01 & -5 \\ 0 & 0 & -5 & 15-i0.01 \end{bmatrix}\]
\[B=\begin{bmatrix} 34 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
\[A^P=\begin{bmatrix} 10 & -5 & 0 & 0 \\ -5 & 15 & -5 & 0 \\ 0 & -5 & 15 & -5 \\ 0 & 0 & -5 & 15 \\ -0.01 & 0 & 0 & 0 \\ 0 & -0.01 & 0 & 0 \\ 0 & 0 & -0.01 & 0 \\ 0 & 0 & 0 & -0.01 \end{bmatrix}\]
\[B^P=\begin{bmatrix} 34 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
Let's assume we want to preserve registers 0-4. Thus we start with register 5 for matrix AP.
Enter matrix AP:
P.FCN:CLREGS
5.0804
XEQ 'MED'
Calculate offset for element (1,5):
1
5
5.1604
MATRIX:M.REG
21
Calculate offset for element (1,9):
1
9
5.1604
MATRIX:M.REG
37
Copy complex 4×4 matrix:
21.0404
37
MATRIX:M.COPY
37.0404
Negate 4×4 matrix (-x = x - 2x):
-2
21.0404
37.0404
MATRIX:M+x
Calculate offset for element (1,1):
1
1
5.1604
MATRIX:M.REG
5
Calculate offset for element (1,13):
1
13
5.1604
MATRIX:M.REG
53
Copy real 4×4 matrix:
5.0404
53
MATRIX:M.COPY
53.0404
Transpose upper 8×4 matrix:
5.0804
MATRIX:TRANSP
5.0408
Transpose lower 8×4 matrix:
37.0804
MATRIX:TRANSP
37.0408
Transpose 8×8 matrix:
5.0808
MATRIX:TRANSP
5.0808
Enter matrix BP:
70.0801
XEQ 'MED'
Invert matrix A:
5.0808
MATRIX:M-1
Multiply A by B:
5.0808
70.0801
80
MATRIX:M×
80.0801
Transpose 2×4 matrix:
80.0204
MATRIX:TRANSP
80.0402
Result:
\[I=\begin{bmatrix} 4.2000 & 0.0061 \\ 1.600 & 0.0037 \\ 0.6000 & 0.0019 \\ 0.2000 & 0.0008 \end{bmatrix}\]
This article describes how to perform matrix multiplication and matrix inversion with complex matrices. Since the WP-34S lacks complex numbers the same trick as in the HP-15C is used. The complex transformation between \(Z^P\) and \(\tilde{Z}\) is provided by combining M.COPY and M.TRANSP commands. Of course this could be written down in a small program.
MED: Matrix Editor
This description is from matrixedit.wp34s:
Code:
/*
* Interactive editor for matrices.
* Start the editor with a matrix descriptor in X.
* Use the arrow keys to navigate:
* up/down: previous/next column
* f-up/f-down: previous/next row
* Type a digit to enter a number. This will suspend
* the progrom. R/S will enter X at the current row/column.
* XEQ RCL restores the present value to X if you decide
* not to change the cell. Press R/S to continue.
*
* Some hot keys:
* A will start over at (1,1).
* When used outside the editor you will need to specifiy
* a new descriptor first.
* B (labeled 1/x) will call M[^-1] on the matrix.
* C continues at the last position.
* D computes and displays the determinant. R/S or C continue.
*
* We need a few registers for internal use and therefore
* set the stacksize to 4 to free registers A to D.
* Press <- to restore the mode and exit the editor.
* Double precision mode will always be reset!
*
* Register usage:
* A - Matrix descriptor
* B - Current register
* D - Old setup mode
* I, J - Row and column
* K last key pressed
*
* Flags:
* A - Controls big "=" sign
* 00 - Shift state
*
* Fixed labels:
* A, 11 - Start over with new matrix ('A'gain)
* B, 12 - Compute the inverse ('1/x')
* C - Continue with current matrix ('C'ontinue)
* 14, D - Compute the determinant ('D'eterminant)
* 22 - [RCL] recalls current matrix element
* 24 - [f] toggles navigation from horizontal to vertical
* 31 - [ENTER^] (same as recall)
* 35 - [<-] Backspace to exit the editor
* 51 - [^] Navigation up/left
* 61 - [v] Navigation down/right
*/HP-15C
Owner's Handbook
Section 12: Calculating With Matrices
Calculating With Complex Matrices
Inverting a Complex Matrix
Example: (pp. 165)
\[Z = \begin{bmatrix} 4+3i & 7-2i \\ 1+5i & 3+8i \end{bmatrix}\]
\[Z^P = \begin{bmatrix}4 & 7 \\ 1 & 3 \\ 3 & -2 \\ 5 & 8\end{bmatrix}\]
Enter matrix \(Z^P\):
0.0402
XEQ 'MED'
4 R/S
↓ 7 R/S
↓ 1 R/S
(…)
↓ 8 R/S
Calculate offset for element (1,3):
1
3
0.0802
MATRIX:M.REG
4
Calculate offset for element (1,5):
1
5
0.0802
MATRIX:M.REG
8
Copy complex 2×2 matrix:
4.0202
8
MATRIX:M.COPY
8.0202
Negate 2×2 matrix (-x = x - 2x):
-2
4.0202
8.0202
MATRIX:M+x
Calculate offset for element (1,1):
1
1
0.0402
MATRIX:M.REG
0
Calculate offset for element (1,7):
1
7
0.0802
MATRIX:M.REG
12
Copy real 2×2 matrix:
0.0202
12
MATRIX:M.COPY
12.0202
Transpose upper 4×2 matrix:
0.0402
MATRIX:TRANSP
0.0204
Transpose lower 4×2 matrix:
8.0402
MATRIX:TRANSP
8.0202
Transpose 4×4 matrix:
0.0404
MATRIX:TRANSP
0.0404
\[\tilde{Z}=\begin{bmatrix} 4 & 7 & -3 & 2 \\ 1 & 3 & -5 & -8 \\ 3 & -2 & 4 & 7 \\ 5 & 8 & 1 & 3 \end{bmatrix}\]
Calculate the inverse:
0.0404
MATRIX:M-1
\[\tilde{Z}^{-1}=\begin{bmatrix} -0.0254 & 0.2420 & 0.2829 & 0.0022 \\ -0.0122 & -0.1017 & -0.1691 & 0.1315 \\ -0.2829 & -0.0022 & -0.0254 & 0.2420 \\ 0.1691 & -0.1315 & -0.0122 & -0.1017 \end{bmatrix}\]
Solving the Complex Equation AX = B
Example: (pp. 169)
\[A^P=\begin{bmatrix} 10 & 0 \\ 0 & 0 \\ 200 & -200 \\ -200 & 170 \end{bmatrix}\]
\[B^P=\begin{bmatrix} 5 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
Calculate inverse of AP as above and multiply by BP:
0.0404
16.0401
20
MATRIX:M×
20.0401
\[\tilde{A}^{-1}\times B^P=\begin{bmatrix} 0.0372 \\ 0.0437 \\ 0.1311 \\ 0.1543 \end{bmatrix}\]
Transpose 2×2 matrix:
20.0202
MATRIX:TRANSP
\[\begin{bmatrix} 0.0372 & 0.1311 \\ 0.0437 & 0.1543 \end{bmatrix}\]
HP-42S
RPN Scientific
Programming Examples and Techniques
5: Matrices
Solving Simultaneous Equations That Have Complex Terms
Example: (pp. 166)
\[A=\begin{bmatrix} 10-i0.01 & -5 & 0 & 0 \\ -5 & 15-i0.01 & -5 & 0 \\ 0 & -5 & 15-i0.01 & -5 \\ 0 & 0 & -5 & 15-i0.01 \end{bmatrix}\]
\[B=\begin{bmatrix} 34 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
\[A^P=\begin{bmatrix} 10 & -5 & 0 & 0 \\ -5 & 15 & -5 & 0 \\ 0 & -5 & 15 & -5 \\ 0 & 0 & -5 & 15 \\ -0.01 & 0 & 0 & 0 \\ 0 & -0.01 & 0 & 0 \\ 0 & 0 & -0.01 & 0 \\ 0 & 0 & 0 & -0.01 \end{bmatrix}\]
\[B^P=\begin{bmatrix} 34 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\]
Let's assume we want to preserve registers 0-4. Thus we start with register 5 for matrix AP.
Enter matrix AP:
P.FCN:CLREGS
5.0804
XEQ 'MED'
Calculate offset for element (1,5):
1
5
5.1604
MATRIX:M.REG
21
Calculate offset for element (1,9):
1
9
5.1604
MATRIX:M.REG
37
Copy complex 4×4 matrix:
21.0404
37
MATRIX:M.COPY
37.0404
Negate 4×4 matrix (-x = x - 2x):
-2
21.0404
37.0404
MATRIX:M+x
Calculate offset for element (1,1):
1
1
5.1604
MATRIX:M.REG
5
Calculate offset for element (1,13):
1
13
5.1604
MATRIX:M.REG
53
Copy real 4×4 matrix:
5.0404
53
MATRIX:M.COPY
53.0404
Transpose upper 8×4 matrix:
5.0804
MATRIX:TRANSP
5.0408
Transpose lower 8×4 matrix:
37.0804
MATRIX:TRANSP
37.0408
Transpose 8×8 matrix:
5.0808
MATRIX:TRANSP
5.0808
Enter matrix BP:
70.0801
XEQ 'MED'
Invert matrix A:
5.0808
MATRIX:M-1
Multiply A by B:
5.0808
70.0801
80
MATRIX:M×
80.0801
Transpose 2×4 matrix:
80.0204
MATRIX:TRANSP
80.0402
Result:
\[I=\begin{bmatrix} 4.2000 & 0.0061 \\ 1.600 & 0.0037 \\ 0.6000 & 0.0019 \\ 0.2000 & 0.0008 \end{bmatrix}\]