Register to reply 
Probability of a matrix having full rank 
Share this thread: 
#1
Mar1612, 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 NK 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 NK 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, NK < 2^K so the total number of columns I have to make the selection is NK 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. 


Register to reply 
Related Discussions  
Matrix Multiplication and Rank of Matrix  General Math  2  
Obtaining an invertible square matrix from a nonsquare 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 