Question

Let be a matrix with rank . Then there exist invertible matrices and such that

Let A be a let be the nullity of ie,

let be a basis of
extend this basis to a basis of say
is a basis for
Extend this to a basis of
Let be the matrix whose columns are elements of and is the inverse of the matrix whose columns are formed by elements of

Theorem.

Every non null matrix has a rank factorization.

Let where is of order
is of order

Claim.

another proof
Let A be of rank
Let be a basis where each
So, every column of A say is a linear combination of elements in
Let

if then

Theorem.

If is a rank factorization of A, then is also a rank factorization of A for a non-singular matrix.
Also, every rank factorization of is of this form

#10-oct

Exercise.

Determine all projectors of order over

there are 3 kinds of projection

a)
b)
c) restriction of det and trace trace=1 and det =0

Exercise.

If then show rank of A is the trace of A