HP 15c Manual

 Section 12: Calculating with Matrices 161 
 
Instead,  calculations  with  complex  matrices  are  performed  by  using  real 
matrices  derived  from  the  original  complex  matrices – in a  manner  to  be 
described below – and performing certain transformations in addition to the 
regular  matrix  operations.  These  transformations  are  performed  by  four 
calculator functions. This section will describe how to do these calculations. 
(There  are  more  examples  of  calculations  with  complex...

162 Section 12: Calculating with Matrices 
 
Suppose  you  need  to  do  a  calculation  with  a  complex  matrix  that  is  not 
written  as  the  sum  of  a  real  matrix  and  an  imaginary  matrix – as  was  the 
matrix Z in  the  example  above – but  rather  written  with  an  entire  complex 
number in each element, such as 
. 
This  matrix  can  be  represented  in  the  calculator  by  a  real  matrix  that  looks 
very  similar – one  that  is  derived  simply  by  ignoring  the i and  the  +...

 Section 12: Calculating with Matrices 163 
 
Example: Store the complex matrix 
 
in the form ZC, since it is written in a form that shows ZC. Then transform 
ZC into the form ZP. 
You  can  do  this  by  storing  the  elements  of ZC in  matrix A and  then  using 
the p function, where 
 
 
Keystrokes Display  
´> 0  Clears all matrices. 
2 v 4 
´mA 
 4.0000 Dimensions matrix A to be 
2×4. 
´> 1  4.0000 Sets beginning row and 
column numbers in R0 and 
R1 to 1. 
´U  4.0000 Activates User mode. 
4 OA...

164 Section 12: Calculating with Matrices 
 
Matrix A now represents the complex matrix Z in ZP form: 
 
The Complex Transformations Between ZP and Z  
An  additional  transformation  must  be  done  when  you  want  to  calculate  the 
product  of  two  complex  matrices,  and  still  another  when  you  want  to 
calculate  the  inverse  of  a  complex  matrix.  These  transformations  convert 
between  the ZP representation  of  an m×n complex  matrix  and  a 2m×2n 
partitioned matrix of the following...

 Section 12: Calculating with Matrices 165 
 
Inverting a Complex Matrix 
You can calculate the inverse of a complex matrix by using the fact that  
(  )-1 = (  -1). 
 To calculate inverse, Z-1, of a complex matrix Z: 
1. Store the elements of Z in memory, in the form either of ZP or of ZC  
2. Recall the descriptor of the matrix representing Z into the display. 
3. If the elements of Z were entered in the form ZC, press ´p to 
transform ZC into ZP 
4. Press ´ > 2 to transform ZP into   . 
5. Designate...

166 Section 12: Calculating with Matrices 
 
 
Keystrokes Display 
´ < 
B 
A 4 4 Designates B as the result 
matrix. 
∕ b 4 4 Calculates (  )-1 = (  -1) and 
places the result in matrix B. 
´> 3 b 4 2 Transforms (  -1) into  
(  -1)P. 
The representation of Z-1 in partitioned form is contained in matrix B. 
 
Multiplying Complex Matrices 
The product of two complex matrices can be calculated by using the fact 
that (YX)P =   P. 
To calculate YX, where Y and X are complex matrices: 
1. Store  the...

 Section 12: Calculating with Matrices 167 
 
8. Press * to calculate  XP = (YX)P. The values of these matrix 
elements  are  placed  in  the  result  matrix,  and  the  descriptor  of 
the result matrix is placed in the X-register. 
9. If you want the product in the form (YX)C, press |c 
Note that you dont transform XP into   . 
You can derive the complex elements of the matrix product YX by recalling 
the  elements  of  (XY)P or  (YX)C and  combining  them  according  to  the 
conventions described...

168 Section 12: Calculating with Matrices 
 
Writing down the elements of C, 
, 
where  the  upper half of  matrix C is the  real part of ZZ-1 and the  lower  half 
is the imaginary part. Therefore, by inspection of matrix C, 
 
As expected, 
 
Solving the Complex Equation AX = B 
You  can  solve  the  complex  matrix  equation AX = B by  finding X = A-1B. 
Do this by calculating XP = (Ã)-1 BP. 
To solve the equation AX = B, where A, X, and B are complex matrices: 
1. Store  the  elements  of A and B in...

 Section 12: Calculating with Matrices 169 
 
 
4. Recall the descriptor of the matrix representing A into the display. 
5. If  the  elements  of A were  entered  in  the  form  of AC,  press ´ 
p to transform AC into AP. 
6. Press ´> 2 to transform AP into Ã. 
7. Designate  the  result  matrix;  it  must  not  be  the  same  as  the  matrix 
representing A. 
8. Press ÷;  this  calculates XP.  The  values  of  these  matrix  elements 
are placed in the result matrix, and the descriptor of the result...

170 Section 12: Calculating with Matrices 
 
In partitioned form, 
, 
where the zero elements correspond to real and imaginary parts with zero 
value. 
Keystrokes Display 
4 v2´mA  2.0000 Dimensions matrix A to be 
4×2. 
´> 1  2.0000 Set beginning row and column 
numbers in R0 and R1 to 1. 
´U  2.0000 Activates User mode. 
10 OA  10.0000 Stores a11. 
0 O A  0.0000 Stores a12. 
OA  0.0000 Stores a21. 
OA  0.0000 Stores a22. 
200 OA  200.0000 Stores a31. 
“OA –200.0000 Stores a32. 
OA –200.0000 Stores a41....