Maple can handle a wide variety of matrix and
linear algebra operations. The first step is
letting Maple know you will be doing some
linear algebra. Do so with the
with(linalg) command. Maple will
list all of the available commands (No, we
will not be going through all of them in this
exercise!) on the screen:
> with(linalg);
Warning: new definition for norm
Warning: new definition for trace
[BlockDiagonal, GramSchmidt, JordanBlock,Wronskian, add, addcol, addrow, adj,
adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix,
charmat, charpoly, col, coldim, colspace,colspan, companion, concat,
cond, copyinto, crossprod, curl, definite,delcols, delrows, det, diag,
diverge, dotprod, eigenvals, eigenvects,entermatrix, equal, exponential,
extend, ffgausselim, fibonacci, frobenius,gausselim, gaussjord,
genmatrix, grad, hadamard, hermite,hessian, hilbert, htranspose,
ihermite, indexfunc, innerprod, intbasis,inverse, ismith, iszero,
jacobian, jordan, kernel, laplacian,leastsqrs, linsolve, matrix, minor,
minpoly, mulcol, mulrow, multiply, norm,normalize, nullspace, orthog,
permanent, pivot, potential, randmatrix,randvector, rank, ratform, row,
rowdim, rowspace, rowspan, rref,scalarmul, singularvals, smith, stack,
submatrix, subvector, sumbasis, swapcol,swaprow, sylvester, toeplitz,
trace, transpose, vandermonde, vecpotent,vectdim, vector]
(Ignore the two Warnings. They are there to
let us know of new definitions of
norm and trace since
the last version of Maple)
Go ahead and define a matrix using the
array variable:
> A := array([ [1,2,3], [x,x/2,3-x],[2/3, 0, -5/x] ]);
[ 1 2 3 ]
[ ]
A := [ x 1/2 x 3 - x ]
[ ]
[ 2/3 0 - 5/x ]
Notice that we placed each row in square
brackets, []. We also blocked off the
entire array with square brackets,
[].
Now, using evalm, compute the
matrix inverse of A:
> Ainv := evalm(A^(-1));
[ 15 60 - 12 + 7 x ]
[ ----------- ---------------- 3 ----------- ]
[ - 69 + 14 x x (- 69 +14 x) - 69 + 14 x ]
[ ]
[ - 21 + 2 x 5 + 2x - 3 + 4 x ]
Ainv := [ 2 ----------- 6--------------- - 6 ----------- ]
[ - 69 + 14 x x (- 69 +14 x) - 69 + 14 x ]
[ ]
[ x 8 x ]
[ 2 ----------- ------------ 9 ----------- ]
[ - 69 + 14 x - 69 +14 x - 69 + 14 x ]
We know that [A][1/A] should yield identity. Let us confirm that:
> evalm( A &* Ainv );
[ 1 0 0 ]
[ ]
[ 0 1 0 ]
[ ]
[ 0 0 1 ]
What is the determinant of A?
> det(A);
23/2 - 7/3 x
Now go ahead an define a vector, B:
> B:= array(1..3): B[1]:= 9: B[2]:= x;
B[2] := x
> print(B);
[ 9, x, B[3] ]
We can multiply the array and the vector using
the &* to symbolize this
operation:
> A &* B;
A &* B
Remember, Maple stores the answer
symbolically until called for with the
evalm command:
> evalm(");
2 6 x - 5 B[3]
[ 9 + 2 x + 3 B[3], 9 x + 1/2 x + 3B[3] - B[3] x, ------------ ]
x
We will now define another vector, C:
> C:= array(1..3): C[1]:=-x: C[2]:= 12;
C[2] := 12
> print(C);
[ - x, 12, C[3]]
And when we multiply B.C :
> evalm(B &* C); Error, (in linalg[multiply]) vector dimensions incompatible
Uh Oh! A common mistake occurs when
&* is used to represent the dot
product. The dotprod command
fixes this:
> evalm( dotprod(B,C) );
3 x + B[3] C[3]
Much better!
This is just a start. Maple allows for
eigenvalue calculations, exponentials of
matrices and vectors, Hermitian operations,
and a whole host of other linear algebra
tools. Look at one of the resources
for a more detailed explanation of linear
algebra tools available with Maple.
L O S A L A M O S N A T I O N A L L A B O R A T O R Y
Operated by the University of California for the U.S. Department of Energy