Register to reply

Probability of a matrix having full rank

by anu914
Tags: combination, matrices, probabality
Share this thread:
anu914
#1
Mar16-12, 12:30 AM
P: 3
Hi all,

I am trying to find the probability that a matrix has full rank.

Consider a K*N matrix where the first K columns are linearly independent columns and the next N-K columns are linear combinations of these K columns.

I want to find the probability that a sub matrix formed by randomly selecting columns of this matrix has full rank. (or all the columns of this sub matrix are linearly independent).

My logic is as follows,

Step 1 : Select u1 number of columns randomly from the first K columns. Then rank(Gu) = u1.
No. of ways to select = K choose u1

Step 2: Now I select one column from the N-K columns and check whether this belong to the span of u1 columns. If not then I increase rank by one.
span of u1 contain 2^u1 possibilities.
So ideally I have to select 1 from 2^K - 2^u1 columns in order to have rank(Gu) = u1 + 1

But my problem is that, N-K < 2^K so the total number of columns I have to make the selection is N-K and not 2^K.

I'm finding it really difficult to interpret this in mathematical formulas using combinations.

Really appreciate if someone can help.

Thanks in advance.
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker

Register to reply

Related Discussions
Matrix Multiplication and Rank of Matrix General Math 2
Obtaining an invertible square matrix from a non-square matrix of full rank Linear & Abstract Algebra 0
Rank of a Matrix Calculus & Beyond Homework 1
Help with full rank factorization Linear & Abstract Algebra 0
Matrix manipulations/rank of a matrix Calculus & Beyond Homework 2