HP 15c Manual

							 Section 12: Calculating with Matrices 151 
 
Example: Calculate the transpose of matrix B. Matrix B was set in 
preceding examples to 
 
 
Keystrokes Display 
l > B b 2 3 Displays descriptor of 
2×3 matrix B. 
´ > 4  b 3 2 Descriptor of 3×2 
transpose. 
Matrix B (which you can view using l B in User mode) is now 
 
Scalar Operations 
Scalar  operations  perform  arithmetic  operations  between  a  scalar  (that  is,  a 
number)  and  each  element  of  a  matrix.  The  scalar  and  the  descriptor  of  the 
matrix must be placed in the X- and Y-registers – in either order. (Note that 
the register position will affect the outcome of the - and ÷ functions.) 
The  resulting  values  are  stored  in  the  corresponding  elements  of  the  result 
matrix. 
The possible operations are shown in the following table..954
321

B .
93
52
41






B  
						

							152 Section 12: Calculating with Matrices 
 
 
Operation 
Elements of Result Matrix* 
Matrix in Y-Register Scalar in Y-Register 
Scalar in X-Register Matrix in X-Register 
+ Adds scalar value to each matrix element. 
* Multiplies each matrix element by scalar value. 
- Subtracts scalar value 
from each matrix 
element. 
Subtracts each matrix 
element from scalar value. 
÷ Divides each matrix 
element by scalar value. 
Calculates inverse of matrix 
and multiplies each element 
by scalar value. 
* Result matrix may be the specified matrix. 
Example: Calculate the matrix B = 2A. then subtract 1 from every element 
in B. From before, use 
 
. 
Keystrokes Display 
´ A A 2 3 Displays descriptor of matrix A. 
2 * b 2 3 Redimensions matrix B to the 
same dimensions as A, multiplies 
the elements of A by 2, stores 
those values in the corresponding 
elements of B, and displays the 
descriptor of the result matrix. 

954
321A  
						

							 Section 12: Calculating with Matrices 153 
 
 
Keystrokes Display 
1 - b 2 3 Subtracts 1 from the elements of 
matrix B and stores those values in the 
same elements of B. 
 
The result (which you can view using lB in User mode) is 
. 
Arithmetic Operations 
With matrix descriptors in both the X- and Y-registers, pressing + or 
- calculates the sum or difference of the matrices. 
Pressing Calculates* 
+ Y + X 
- Y - X 
* Result is stored in result matrix. 
Result matrix may be X or Y 
Example: Calculate C = B - A, where A and B are defined in the previous 
example. 
 
Keystrokes Display 
´< C    Designates C as result matrix. 
l> B b 2 3 Recalls descriptor of matrix B. 
(This step can be skipped if 
descriptor is already in X-register.) 
l> A A 2 3 Recalls descriptor of matrix A into 
X-register, moving descriptor of 
matrix B to Y-register. 


1797
531B . and  1797
531
954
321







BA  
						

							154 Section 12: Calculating with Matrices 
 
 
Keystrokes 
 
Display 
   
- C 2 3 Calculates B - A and stores 
values in redimensioned result 
matrix C. 
The result is  
Matrix Multiplication 
With  matrix  description  in  both  the  X- and  Y-registers,  you  can  calculate 
three  different  matrix  products.  The  table  below  shows  the  results  of  the 
three  functions  for  a  matrix X specified  in  the X-register  and  a  matrix Y 
specified  in  the  Y-register.  The  matrix X-1 is  the  inverse  of X,  and  the 
matrix YT is the transpose of Y. 
 
Pressing Calculates* 
* YX 
´ > 5 YTX 
÷ X-1Y 
*  Result  is  stored  in  result  matrix.  For ÷,  the 
result matrix can be Y but not X. For the others, 
the result matrix must be other than X or Y. 
Note: When  you  use  the ÷ function  to  evaluate  the  expression 
A-1B, you must enter the matrix descriptors in the order B, A rather 
than in the order that they appear in the expression.* 
The  value  stored  in  each  element  of  the  result  matrix  is  determined 
according to the usual rules of matrix multiplication. 
For > 5, the  matrix  specified  in  the  Y-register  isnt  changed  by  this 
operation,  even  though  its  transpose  is  used.  The  result  is  identical  to  that 
obtained using > 4 (transpose) and *. 
                                                           * This is the same order you would use if you were entering b and a for evaluating a-1b = b/a 

843
210C  
						

							 Section 12: Calculating with Matrices 155 
 
For ÷,  the  matrix  specified  in  the  X-register  is  replaced  by  its LU 
decomposition.  The ÷ function  calculates X–1Y using  a  more  direct 
method than does ∕ and *, giving the  result faster and  with improved 
accuracy. 
Example: Using  matrices A and B from  the  previous  example,  calculate 
C = AT B. 
 
Keystrokes Display 
l>
A 
A 2 3 Recalls descriptor for matrix A. 
l>
B 
b 2 3 Recalls descriptor for matrix B 
into X-register, moving matrix 
A descriptor into Y-register. 
´< 
C 
b 2 3 Designates matrix C as result 
matrix. 
´> 5 C 3 3 Calculates AT B and stores 
result in matrix C, which is 
redimensioned to 3×3. 
The result, matrix C, is 
. 
 






1797
531
954
321  and  BA 






1689066
955137
733929
C  
						

							156 Section 12: Calculating with Matrices 
 
Solving the Equation AX = B 
The ÷ function is useful for solving 
matrix  equations  of  the  form AX  =  B, 
where A is the  coefficient  matrix, B is 
the  constant  matrix,  and X is  the 
solution  matrix.  The  descriptor  of  the 
constant matrix B should be entered in 
the Y-register and the descriptor of the 
coefficient matrix A should be entered 
in  the  X-register  Pressing ÷ then 
calculates the solution X=A-1B.* 
Remember  that  the ÷ function  replaces  the  coefficient  matrix  by  its LU 
decomposition  and  that  this  matrix  must  not  be  specified  as  the  result 
matrix.  Furthermore,  using ÷ rather  than ∕ and * gives  a  solution 
faster and with improved accuracy. 
At  the  beginning  of  this  section,  you  found  the  solution  for  a  system  of 
linear  equations  in  which  the  constant  matrix  and  the  solution  matrix  each 
had one column. The following example illustrates that you can use the HP-
15C  to  find  solutions  for  more  than  one  set  of  constants—that  is,  for  a 
constant matrix and solution matrix with more than one column. 
 
Example: Looking at his receipts for his 
last  three  deliveries  of  cabbage  and 
broccoli, Silas Farmer sees the following 
summary. 
                                                           * If A is a singular matrix (that is, one that doesn’t have an inverse), then the HP-15C modifies the LU form of  A  by  an  amount  that  is  usually  small  compared  to  round-off  error.  The  calculated  solution  corresponds to that for a nonsingular coefficient matrix close to the original, singular matrix. 
Y constant matrix 
X coefficient 
matrix 
   
						

							 Section 12: Calculating with Matrices 157 
 
 
 Week 
1 2 3 
Total Weight (kg) 274 233 331 
Total Value $120.32 $112.96 $151.36 
Silas  knows  that  he  received  $0.24 per  kilogram  for  his  cabbage  and  $0.86 
per  kilogram  for  his  broccoli.  Use  matrix  operations  to  determine  the 
weights of cabbage and broccoli he delivered each week. 
Solution: Each  weeks  delivery  represents  two  linear  equations  (one  for 
weight  and  one  for  value)  with  two  unknown  variables  (the  weights  of 
cabbage and broccoli). All three weeks can be handled simultaneously using 
the matrix equation 
  =  
or     AD = B 
where the first row of matrix D is the weights of cabbage for the three 
weeks and the second row is the weights of broccoli. 
Keystrokes Display  
2 
v´mA 
2.0000 Dimensions A as 2×2 matrix. 
´> 1 2.0000 Sets row and column numbers in R0 
and R1 to 1. 
´U 2.0000 Activates User mode. 
1 OA 1.0000 Stores a11. 
OA 1.0000 Stores a12. 
.24 OA 0.2400 Stores a21. 
.86 OA 0.8600 Stores a22. 
2 v 3 
´mB 
3.0000 Dimensions B as 2×3 matrix. 


0.860.24
11 


232221
1312
ddd
ddd11 


151.36112.96120.32
331233274  
						

							158 Section 12: Calculating with Matrices 
 
 
Keystrokes Display  
274 OB 274.0000 Stores b11.* 
233 OB 233.0000 Stores b12. 
331 OB 331.0000 Stores b13. 
120.32 OB 120.3200 Stores b21. 
112.96 OB 112.9600 Stores b22. 
151.36 OB 151.3600 Stores b23. 
´< Á 151.3600 Designates matrix D as result 
matrix. 
l> B b 2 3 Recalls descriptor of constant 
matrix. 
l> A A 2 2 Recalls descriptor of coefficient 
matrix A into X-register, moving 
descriptor of constant matrix B 
into Y-register. 
÷ d 2 3 Calculates A-1B and stores result 
in matrix D. 
lÁ 186.0000 Recalls d11, the weight of cabbage 
for the first week. 
lÁ 141.0000 Recalls d12 the weight of cabbage 
for the second week. 
lÁ 215.0000 Recalls d13. 
lÁ 88.0000 Recalls d21. 
lÁ 92.0000 Recalls d22. 
lÁ 116.0000 Recalls d23. 
´U 116.0000 Deactivates User mode. 
                                                           * Note that you did not need to press ´> 1 before beginning to store the elements of matrix B. This is  because  after  you  stored  the  last  element  of  matrix A,  the  row  and  column  numbers  in  R0 and  R1 were automatically reset to 1.  
						

							 Section 12: Calculating with Matrices 159 
 
Silas deliveries were: 
 Week 
 1 2 3 
Cabbage (kg) 186 141 215 
Broccoli (kg) 88 92 116 
Calculating the Residual 
The HP-15C enables you to calculate the residual, that is, the matrix 
Residual = R–YX 
where R is  the  result  matrix  and X and Y are  the  matrices  specified  in  the 
X- and Y-registers. 
This  capability  is  useful,  for  example,  in  doing  iterative  refinement  on  the 
solution  of  a  system  of  equations  and  for  linear  regression  problems.  For 
example, if C is a possible solution for AX = B, then B – AC indicates how 
well  this  solution  satisfies  the  equation.  (Refer  to  the HP-15C  Advanced 
Functions  Handbook for  information  about  iterative  refinement  and  linear 
regression.) 
The  residual  function  (> 6)  uses  the  current  contents  of  the  result 
matrix  and  the  matrices  specified  in  the X- and Y-registers  to  calculate  the 
residual defined above. The residual is stored in the result matrix, replacing 
the original result matrix. A matrix specified in the X- or Y-register can not 
be the result matrix. 
Using > 6  rather  than * and - gives  a  result  with  improved 
accuracy, particularly if the residual is small compared to the matrices being 
subtracted. 
To calculate the residual: 
1. Enter the descriptor of the Y matrix into the Y-register. 
2. Enter the descriptor of the X matrix into the X-register. 
3. Designate the R matrix as the result matrix. 
4. Press ´> 6. The residual replaces the original result 
matrix (R). The descriptor of the result matrix is placed in the X-
register.  
						

							160 Section 12: Calculating with Matrices 
 
Using Matrices in LU Form 
As noted  earlier,  two  matrix  operations  (calculating  a  determinant  and 
solving  the  matrix  equation  (AX  =  B)  create  an LU decomposition  of  the 
matrix  specified  in  the  X-register.  The  descriptor  of  such  a  matrix  has  two 
dashes  following  the matrix  name. A matrix  in LU form  has  elements  that 
differ from the elements of the original matrix. 
However, the descriptor for a matrix in LU form can be used in place of the 
descriptor for the original  matrix for operations involving the inverse of the 
matrix  and  for  the  determinant  operation. That  is,  either  the  original  matrix 
or its LU decomposition can be used for these operations: 
∕  
÷ for the matrix in the X-register 
> 9 
For  these  three  functions,  using  the LU form  of  the  matrix  to  be  inverted 
gives a result that is identical to that using the original matrix. 
As an example, if you solved the  matrix equation AX  = B, matrix A would 
be changed to its LU form. If you wanted to change  the B matrix and solve 
the equation again, you could do so without changing the A matrix – the LU 
matrix will give the correct solution. 
For all other matrix operations, a  matrix that is an LU decomposition is not 
recognized  as  representing  its  original  matrix.  Instead,  the  elements  of  the 
LU matrix  are  used  just  as  they  appear  in  matrix  memory  and  the  result  is 
not the result you would obtain using the original matrix. 
Calculations With Complex Matrices 
The  HP-15C  enables  you  to  perform  matrix  multiplication  and  matrix 
inversion  with  complex  matrices  (that  is,  matrices  whose  elements  are 
complex  numbers)  and  to  solve  systems  of  complex  equations  (that  is, 
equations whose coefficients and variables are complex). 
However,  the  HP-15C  stores  and  operates  on  only  real  matrices.  The 
capability  of  doing  calculations  with  complex  matrices  is  completely 
independent  of  the  capability  of  doing  calculations  with  complex  numbers 
described  in  the  preceding  section.  You  don’t need  to  activate  Complex 
mode for calculations with complex matrices.