Page 817 - Advanced_Engineering_Mathematics o'neil

APPENDIX A A MAPLE Primer
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Multiply matrices A and B by
A.B;
Elementary row operations can be carried out very efficiently. Let A be an n × m matrix that
has been entered. To interchange rows k and j and A to form a new matrix B, use
B:=swaprow(A,k,j);
To form S from A by multiplying row r of B by x, use
S:=mulrow(A,r,x);
And to form T from A by adding x times row r 1 to row r 2 ,use
T:= addrow(A,r1,r2,x);
Finally, suppose we select an element a k, j of A, and we want to proceed by elementary row
operations to a new matrix W having zeros above and below this element. This is called pivoting
about a k, j , and is done using the pivot command. Enter
W:=pivot(A,k,j);
W will be the matrix with zeros above and below a k, j , which could have been obtained by
systematically applying row operations as needed to each of the rows above and below row k of
the matrix.
Using the pivot command, it is easy to find the reduced row echelon form of a matrix.
Starting in the upper left corner of the matrix, pivot about the leading element of each row. The
resulting matrix is nearly reduced. The only problem is that leading entries of nonzero rows may
not equal 1. Just divide each nonzero row by its leading entry (using mulrow) to obtain leading
entries of 1, resulting in a reduced row echelon matrix.
The corresponding column operations can be achieved by replacing row with col in these
commands. Such column operations may be useful, for example, in manipulating determinants.
The elementary row and column operations of adding a scalar multiple of one row (column)
to another are also useful in carrying out an LU factorization of matrix. The algorithms for
forming the two matrices in such a factorization depend on these operations.
The characteristic polynomial of a square matrix A is obtained using
charpoly(A,x);
in which the variable in which the polynomial will be written (here x) must be specified. The
command
eigenvals(A);
lists the eigenvalues of A (real or complex), giving each eigenvalue according to its multiplicity.
Note that the imaginary unit i is denoted I in the program. The command
eigenvects(A);
lists each eigenvalue, along with its multiplicity and for each eigenvalue, as many linearly
independent eigenvectors as can be found for that eigenvalue. To illustrate, let
A:=Matrix(3,3,[[1,-1,0],[0,1,1],[0,0,-1]]);
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