User:Trieu/Method of matrix rank

Suppose we want to write program to find out the rank of a matrix A_rxc, where 'r' and 'c' are the number of rows and columns of matrix A.

By any available programming languages, one should be able to define an 2 dimensions array, say A[1->r,1->c] to keep all elements of matrix A_rxc that we are going to find a rank.

Here, the method merely does similar to the process that everyone often do to work out the rank by hand.

function make(i,j,a); // assign value "a" to A[i,j].

if (a==0) {
for (m=j->c) {
if (j==1) {
A[i,m]=A[i,m]+(-A[i,m]*A[1,m]);
} // end of (j==1)
if (j!=1) {
A[i,m]=A[i,m]+(-A[i,m]*A[i-1,m]);
} // end of (j!=1)
} // end for
} // end if
if (a==1) {
for (m=j->c) {
A[i,m]=A[i,m]/A[i,j];
} // end for
} // end if
} // end function
for (j=1->c) {
for (i=j->r) {
if (i==j) { make(i,j,1); }
if (i!=j) { make(i,j,0); }
}
}

Then, A_rxc is now rewritten in echelon form!

We count the non-zero rows by writting such as:

zero=true;
count=0;
for (i=1->r) {
for (j=1->c) {
if (A[i,j]!=0) { zero=false; }
if (zero==true) { count++; }
zero=true;
}
}

Notes: The above illustration does not conform to any programming language, one should match the algorithm appropriate with the language you are using.