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  matrices  in  the  
HP-15C Advanced Functions Handbook.) 
Storing the Elements of a Complex Matrix 
Consider an m×n complex matrix Z = X + iY, where X and Y are real  
m×n matrices. This matrix can be represented in the calculator as a  
2m×n ―partitioned‖ matrix: 
 
The  superscript P signifies  that  the  complex  matrix  is  represented  by  a 
partitioned matrix. 
All  of  the  elements  of ZP are  real  numbers – those  in  the  upper  half 
represent  the  elements  of  the  real  part  (matrix X),  those  in  the  lower  half 
represent the elements of the imaginary part (matrix Y). The elements of ZP 
are  stored in one  of the  five  matrices (A, for example) in the  usual  manner, 
as described earlier in this section. 
For example, if Z = X + iY, where 
 
then Z can be represented in the calculator by 
. P artImaginary 
P art Real
}
}
Y
X

PZ ,     and  2221
1211
2221
1211



yy
yy
xx
xxYX 








2221
1211
2221
1211
yy
yy
xx
xx
P
Y
XZA  
						

							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  +  sign. 
The  2 × 2  matrix Z shown  above,  for  example,  can  be  represented  in  the 
calculator in ―complex‖ form by the 2 × 4 matrix. 
. 
The  superscript C signifies  that  the  complex  matrix  is  represented  in  a 
complex-like form. 
Although a  complex  matrix can be initially represented in the  calculator by 
a matrix of the form shown for ZC, the transformations used for multiplying 
and inverting a  complex  matrix presume  that  the  matrix is represented by a 
matrix of the form shown for ZP. The HP-15C provides two transformations 
that convert the representation of a complex matrix between ZC and ZP: 
Pressing Transforms Into 
´p ZC ZP 
| c ZP ZC 
To do either of these transformations, recall the descriptor of ZC or ZP into 
the display, then press the keys shown above. The transformation is done to 
the specified matrix; the result matrix is not affected. 



22222121
12121111
iyxiyx
iyxiyxZ 



22222121
12121111
yxyx
yxyxCZA  
						

							 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  4.0000 Stores a11. 
3 OA  3.0000 Stores a12. 
7 OA  7.0000 Stores a13. 
2 “ OA -2.0000 Stores a14. 
1 OA  1.0000 Stores a21. 
5 OA  5.0000 Stores a22. 
3 OA  3.0000 Stores a23. 
8 OA  8.0000 Stores a24. 
´U  8 0000 Deactivates User mode. 
l> A  A 2 4 Display descriptor of 
matrix A. 
´ p  A 4 2 Transforms ZC into ZP and 
redimensions matrix A. 
 



ii
ii
8351
2734Z .8351
2734



cZA  
						

							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 form: 
. 
The  matrix    created  by  the > 2  transformation  has  twice  as  many 
elements as ZP. 
For example, the matrices below show how    is related to ZP. 
 
The  transformations that  convert  the  representation  of  a  complex  matrix 
between ZP and    are shown in the following table. 
Pressing Transforms Into 
´ > 2 ZP    
´ > 3    ZP 
To  do  either  of  these  transformations,  recall  the  descriptor  of ZP or    into 
the display, then press the keys shown above. The transformation is done to 
the specified matrix; the result matrix is not affected. P artImagi nary 
P art Real
.
85
23
31
74
}
}






PZA 

XY
YXZ 







6154
5461~
54
61ZZP  
						

							 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  a  matrix  as  the  result  matrix.  It  may  be  the  same  as  the 
matrix in which    is stored. 
6. Press ∕.  This  calculates  (  )-1,  which  is  equal  to  (  -1).  The  values 
of  these  matrix  elements  are  stored  in  the  result  matrix,  and  the 
descriptor of the result matrix is placed in the X-register. 
7. Press ´ > 3 to transform (  -1) into (Z-1)P. 
8. If you want the inverse in the form (Z-1)C, press | c 
You can derive the complex elements of Z-1 by recalling the elements of ZP 
or ZC and then combining them as described earlier. 
Example: Calculate the inverse of the complex matrix Z from the previous 
example. 
. 
Keystrokes Display 
l>A A 4 2 Recalls descriptor of matrix A. 
´ > 2 A 4 4 Transforms ZP into    and 
redimensions matrix A. 






85
23
31
74
PZA  
						

							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  elements of Y and X in  memory, in the  form either of 
ZP or ZC. 
2. Recall  the  descriptor  of  the  matrix  representing Y into  the 
display. 
3. If  the  elements  of Y were  entered  in  the  form of YC,  press 
´p to transform YC into YP. 
4. Press ´> 2 to transform YP into  . 
5. Recall  the  descriptor  of  the  matrix  representing X into  the 
display. 
6. If  the  elements  of X were  entered  in  the  form XC,  press 
´p to transform XC into XP. 
7. Designate  the  result  matrix;  it  must  not  be  the  same  matrix  as 
either of the other two. P artImagi nary 
P art Real
1315.01691.0
0022.02829.0
1017.00122.0
2420.00254.0
}
}










B  
						

							 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 earlier. 
Example: Calculate the product ZZ-1, where Z is the complex matrix given 
in the preceding example. 
Since  elements  representing  both  matrices  are  already  stored  (   in A and 
(Z-1)P in B), skip steps 1, 3, 4, and 6. 
Keystrokes Display 
l>A A 4 4 Displays descriptor of matrix A. 
l>B b 4 2 Displays descriptor of matrix 
B. 
´
						

							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  memory,  in  the  form  either  of ZP or 
of ZC. 
2. Recall the descriptor of the matrix representing B into the display. 
3. If  the  elements  of B were  entered  in  the  form BC,  press ´p to 
transform BC into BP. P1
1011
1011
11
10
100500.1100000.1
108000.3100000.1
0000.1100000.4
108500.20000.1
















ZZC 















1011
1111
11
101
100500.1100000.1
108000.3100000.1
0000.1100000.4
108500.20000.1
i
ZZ 



00
00
10
011 i-ZZ  
						

							 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 matrix 
is placed in the X-register. 
9. If you want the solution in the form XC, press |c. 
Note that you dont transform BP into   . 
You  can  derive  the  complex  elements  of  the  solution X by  recalling  the 
elements  of XP or XC and  combining  them  according  to  the  conventions 
described earlier. 
Example: Engineering  student  A.  C.  Dimmer  wants  to  analyze  the 
electrical  circuit  shown  below.  The  impedances  of  the  components  are 
indicated  in  complex  form.  Determine  the  complex  representation  of  the 
currents I1 and I2. 
 
This system can be represented by the complex matrix equation 
 
 
or AX = B. 
  








0
5
30)(200200    
20020010
2I
I
ii
ii1  
						

							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. 
170 OA  170.0000 Stores a42. 
4 v 1´m 
B 
 1.0000 Dimensions matrix B to be 
4×1. 
0 O>B  0.0000 Stores value 0 in all elements 
of B. 
5 v 1 v  1.0000 Specifies value 5 for row 1, 
column 1. 
O|B  5.0000 Stores value 5 in b11. 
l> B  b 4 1 Recalls descriptor for matrix 
B. 
l> A  A 4 2 Places descriptor for matrix A 
into X-register, moving 
descriptor for matrix B into Y-
register. 
 














0
0
0
5
 and 
170200
200200
00
010
BA