matMulRect¶

Computes the transpose of a matrix, a matrix-vector product, a matrix-matrix product, the bilinear form, or any triple product.

Synopsis¶

matMulRect (string)

Required Arguments¶

char string (Input): String indicating matrix multiplication to be performed.

Return Value¶

The result of the multiplication. This is always an array, even if the result is a single number. If no answer was computed, then None is returned.

Optional Arguments¶

aMatrix, float a[] (Input): The nrowa ×ncola matrix A.
bMatrix, float b[] (Input): The nrowb × ncolb matrix A.
xVector, float x (Input): The vector x of size nx.
yVector, float y (Input): The vector y of size ny.

Description¶

This function computes a matrix-vector product, a matrix-matrix product, a bilinear form of a matrix, or a triple product according to the specification given by string. For example, if “A × x” is given, Ax is computed. In string, the matrices A and B and the vectors x and y can be used. Any of these four names can be used with trans, indicating transpose. The vectors x and y are treated as n × 1 matrices.

If string contains only one item, such as “x” or “trans(A)”, then a copy of the array, or its transpose, is returned. If string contains one multiplication, such as “A × x” or “B × A”, then the indicated product is returned. Some other legal values for string are “trans(y) × A”, “A × trans(B)”, “x × trans(y)”, or “trans(x) × y”.

The matrices and/or vectors referred to in string must be given as optional arguments. If string is “B × x”, then bMatrix and xVector must be given.

Example¶

Let

$\begin{split}A = \begin{bmatrix} 1 & 2 & 9 \\ 5 & 4 & 7 \\ \end{bmatrix} \phantom{...} B = \begin{bmatrix} 3 & 2 \\ 7 & 4 \\ 9 & 1 \\ \end{bmatrix} \phantom{...} x = \begin{bmatrix} 7 \\ 2 \\ 1 \\ \end{bmatrix} \phantom{...} y = \begin{bmatrix} 3 \\ 4 \\ 2 \\ \end{bmatrix}\end{split}$

The arrays $A^T$ , Ax, $x^T A^T$ , AB, $B^T A^T$ , $x^Ty$ , $xy^T$ , and $x^TAy$ are computed and printed.

from pyimsl.math.matMulRect import matMulRect
from pyimsl.math.writeMatrix import writeMatrix

a = [[1, 2, 9], [5, 4, 7]]
b = [[3, 2], [7, 4], [9, 1]]
x = [7, 2, 1]
y = [3, 4, 2]

ans1 = matMulRect("trans(A)", aMatrix=a)
writeMatrix("trans(A)", ans1)

ans = matMulRect("A*x", aMatrix=a, xVector=x)
writeMatrix("A*x", ans)

ans = matMulRect("trans(x)*trans(A)", aMatrix=a, xVector=x)
writeMatrix("trans(x)*trans(A)", ans)

ans = matMulRect("A*B", aMatrix=a, bMatrix=b)
writeMatrix("A*B", ans)

ans = matMulRect("trans(B)*trans(A)", aMatrix=a, bMatrix=b)
writeMatrix("trans(B)*trans(A)", ans)

ans = matMulRect("trans(x)*y", xVector=x, yVector=y)
writeMatrix("trans(x)*y", ans)

ans = matMulRect("x*trans(y)", xVector=x, yVector=y)
writeMatrix("x*trans(y)", ans)

shortx = [7, 2]   # Use only the first two components of x
ans = matMulRect("trans(x)*A*y", aMatrix=a, xVector=shortx, yVector=y)
writeMatrix("trans(x)*A*y", ans)

Output¶

 
         trans(A)
             1            2
1            1            5
2            2            4
3            9            7
 
           A*x
          1            2
         20           50
 
    trans(x)*trans(A)
          1            2
         20           50
 
            A*B
             1            2
1           98           19
2          106           33
 
     trans(B)*trans(A)
             1            2
1           98          106
2           19           33
 
trans(x)*y
         31
 
               x*trans(y)
             1            2            3
1           21           28           14
2            6            8            4
3            3            4            2
 
trans(x)*A*y
         293