I have a matrix A of order n×n. Now I need another matrix B whose (i,j)th entry is detA(i,j), where detA(i,j) denote the determinant of the sub matrix of A formed by deleting ith and jth rows and columns. I'm unable to generate the submatrices A(i,j) for every element at a time.
Let us fix a matrix A, so that the answer can be recovered every time.
Since big coefficients should not occur on such a post, let us work over
F = GF(5) first. We start with the matrix:
F = GF(5) A = matrix( F, 5, 5, [ 1 + k^2 + n^3 for k in F for n in F ] )
This is:
sage: A
[1 2 0 0 2]
[2 3 1 1 3]
[0 1 4 4 1]
[0 1 4 4 1]
[2 3 1 1 3]Now, if the (pythonically numbered) rows 1,2, and the columns 0,4 have to be deleted, we have to extract (complementarily) the rows 0,3,4, and the columns 1,2,3. This is done for instance via:
sage: A[ [0,3,4] , [1,2,3] ]
[2 0 0]
[1 4 4]
[3 1 1]For our purposes, it is better to use the form:
R = range( 5 )
A[ [ k for k in R if k not in [1,2] ] ,
[ k for k in R if k not in [0,4] ] ]This is the same.
And now the answer to your question:
R = range( 5 )
S = set( R )
F = GF(5)
A = matrix( F, 5, 5, [ 1+k^2+n^3 for k in F for n in F ] )
B = matrix( F,
5, 5,
[ A[ list( S.difference( {j,k}) ) ,
list( S.difference( {j,k}) ) ].det()
for j in R
for k in R ] )This is:
sage: B
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]Thank you.
You can create lists of rows and columns and use them to cut out a submatrix. E.g.
sage: M=matrix([[1,2,3,4],[3,4,5,6],[6,7,8,9]])
sage: M[[0,2],[0,2]]
[1 3]
[6 8]For large matrix it is not that easy to construct each submatrix one by one. I need n2+n submatrices for a matrix of order n. I'm looking for a general rule or a program.
Well, but this is a straightforward Python loop to write. In the loop, create two lists, say, ro, co, using range(), remove i-th (resp. j-th) elements from it, call A[ro,co].det()/A[ro,ro].det() to compute the corr. determinant ratio, store it in B.
Sorry, I could not get your point clearly. Can you please explain it with an example? Consider a random matrix of order 10 and from it find the submatrix by deleting 2nd and 7th rows and columns.
A live example with nested for loops: http://sagecell.sagemath.org/?q=dzgfjy
Thank you. But the following one simply serves that purpose.
N = random_matrix(ZZ, 10,10) M=N.delete_rows([2,7]) P=M.delete_columns([2,7]). show(N) show(P)
My problem is that I'm unable to generate all such submatrix at a time. What I want is to replace each element of the given matrix by the determinant of such submatrix of that particular element.
I can find the submatrix for a particular element in the following way.
N = random_matrix(ZZ, 10,10) M=N.delete_rows([2,7]) P=M.delete_columns([2,7]) show(N) show(P)
My problem is that I'm unable to generate all such submatrix at a time. What I want is to replace each element of the given matrix by the determinant of such submatrix of that particular element. What I tried is the following one. But it shows some error.
L = random_matrix(ZZ, 10,10) R=matrix(QQ, 10) for i in range(10): for j in range(10): if i!=j: B(i,j)=L.delete_rows([i,j]) B(i,j)=B(i,j).delete_columns([i,j]) R[i,j]=B(i,j).det() show(R)
Asked: 7 years ago
Seen: 9,283 times
Last updated: Jul 19 '17
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