I have a N×N general matrix H with rank n(<N).
Is there any way to get a n×n matrix with rank n from H?

For example,

      |1     2     3|
  H = |4     8     6|
      |0     0     1|

has three eigenvalues 0,1,9 and its rank is 2. I want to get a 2×2 matrix with rank 2 which corresponds to the eigenspace sappaned by eigenvectors of 1,9.

CodePudding user response：

We are given a 3x3 matrix H that is known to have rank r < 3:

1 2 3
4 8 6
0 0 1

One can obtain an nxn matrix comprised of the intersection of rows and columns of H that has rank n by computing the reduced row echelon form (RREF) of H (also called the row canonical form).

After doing so, for each of n row indices i there will be a column in the RREF that contains a 1 in row i (i.e., the row having index i) and zeroes in all other rows. It is seen here that the RREF of H is the following.

1 2 0
0 0 1
0 0 0

As column 0 (i.e., the column having index 0) in the RREF has a 1 in row 0 and zeroes in all other rows, and column 2 has a 1 in row 1 and zeroes in all other rows, and no other column has a 1 in one row and zeroes in all other rows, we conclude that:

• H has rank 2; and
• the nxn matrix comprised of elements in H that are in rows 0 and 1 and columns 0 and 2 has rank n.

Here an nxn matrix with rank n is therefore found to be

1 3
4 6

The same procedure is followed regardless of the size of H (which need not be square) and the rank of H need not be known in advance.

CodePudding user response：

Using the RowEchelon.jl package, we can apply the method described in @CarySwoveland's answer pretty easily. (This is not my area of expertise though, so any corrections to it are welcome; specifically, the choice of rows as 1 to number of pivots is an educated guess based on some trials.)

julia> H = [1 2 3 
            4 8 6
            0 0 1];

julia> using RowEchelon

julia> _, pivotcols = rref_with_pivots(H)
([1.0 2.0 0.0; 0.0 0.0 1.0; 0.0 0.0 0.0], [1, 3])

julia> result = H[1:length(pivotcols), pivotcols]
2×2 Matrix{Int64}:
 1  3
 4  6

The package is just a home for code that used to be in Base Julia, so you can even just copy the code if you don't want to add it as a dependency.