MATMULT product, matrix1, matrix2
The command MATMULT performs multiplication on matrices of floating point type. The matrix1 is multiplied with matrix2 and the result stored in product. Since a n x m matrix represents a linear transformation which takes n-dimensional vectors and produces m-dimensional vectors from them, the following conditions must be satisfied by the three matrices supplied to MATMULT:
All matrices must be two-dimensional.
DIMN (matrix1, 2) = DIMN (matrix2, 1)
DIMN (matrix1, 1) = DIMN (product, 1)
DIMN (matrix2, 2) = DIMN (product, 2)
The latter three conditions are obviously satisfied by square matrices.
Multiplication of two matrices means that their effect on a vector is combined into one matrix. The following program demonstrates this on a simple square.
The square x is a list of four vectors. x is first rotated with ROT by 45o, the rotated square is stored in y.
Now this y is squeezed in size by one half with SQZ and stored in z. Lines 240 to 280 perform all this and show the process.
After a keystroke, the matrix ROTSQZ will be created as the product of ROT and SQZ. Again the original square is transformed but this time by ROTSQZ which rotates and squeezes in one go. This is done by lines 300 to 350.
Lines 100 to 220 initialise the matrices and set up the window for drawing.
Due to the design of QL graphics, line 100 can be freely omitted.
At the bottom of the listing are three PROCedures:
MATVEC multiplies a vector with a matrix (ie. the vector is transformed by this matrix) and MATVECS does the same for a list of vectors, just calling MATVEC for each individual vector. MATVEC(S) is written in a dimension independent way, just to show how that can be done; there is no check on the parameters, just to save space.
POLYDRAW draws a closed polygon from a supplied list of two-dimensional points.
100 WINDOW 448,200,32,16 110 SCALE 8,-5,-4: PAPER 0: CLS 120 : 130 DIM ROT(2,2): rc=1/SQRT(2) 140 ROT(1,1)=rc: ROT(1,2)=rc 150 ROT(2,1)=-rc: ROT(2,2)=rc 160 DIM SQZ(2,2): SQZ(1,1)=.5: SQZ(2,2)=.5 170 : 180 DIM x(4,2), y(4,2), z(4,2) 190 x(1,1)=-1: x(1,2)= 1 200 x(2,1)= 1: x(2,2)= 1 210 x(3,1)= 1: x(3,2)=-1 220 x(4,1)=-1: x(4,2)=-1 230 : 240 INK 5: POLYDRAW x 250 MATVECS y,ROT,x 260 INK 3: POLYDRAW y 270 MATVECS z,SQZ,y 280 INK 7: POLYDRAW z 290 : 300 PAUSE: CLS 310 DIM ROTSQZ(2,2) 320 INK 5: POLYDRAW x 330 MATMULT ROTSQZ,ROT,SQZ 340 MATVECS z,ROTSQZ,x 350 INK 7: POLYDRAW z 360 : 370 : 380 DEFine PROCedure MATVECS (vectors2, matrix, vectors1) 390 LOCal i 400 FOR i=1 TO DIMN(vectors1) 410 MATVEC vectors2(i),matrix,vectors1(i) 420 END FOR i 430 END DEFine MATVECS 440 : 450 DEFine PROCedure MATVEC (vector2, matrix, vector1) 460 REMark vector2 = matrix * vector1 470 LOCal i,j 480 FOR i=1 TO DIMN(vector2) 490 vector2(i)=0 500 FOR j=1 TO DIMN(matrix,2) 510 vector2(i)=vector2(i)+matrix(i,j)*vector1(j) 520 END FOR j 530 END FOR i 540 END DEFine MATVEC 550 : 560 DEFine PROCedure POLYDRAW (vectors) 570 LOCal i 580 POINT vectors(1,1),vectors(1,2) 590 FOR i=2 TO DIMN(vectors), 1 600 LINE TO vectors(i,1),vectors(i,2) 610 END FOR i 620 END DEFine POLYDRAW
Normally the product of two matrices A*B is not the same as B*A, however, the matrices ROT and SQZ in the above example are an exception to this rule. Replace line 330 with: 330 MATMULT ROTSQZ,SQZ,ROT and nothing will change.