Generate a large almost diagonal matrix

I want to generate a 1000x1000 matrix where the diagonals are 1 and the extries above the diagonal are -1. How do I do that? Thanks!

Answer 1

[Matt J]

N=10;

A=eye(N)-diag(ones(1,N-1),1)

A = 10×10

1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1

Answer 2

[Matt J]

N=10;
e=ones(N,1);
A=spdiags([e,-e],[0,1],N,N);


full(A)

ans = 10×10

1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 1

Answer 3

[John D'Errico]

This is commonly known as a bidiagonal matrix.

It is far and away best defined using sparse storage, as produced by spdiags, since your matrix has only 2 non-zero elements on every row.

But there would be many ways to create such a matrix. You could build it as a Toeplitz matrix. Of course, that would be full, not sparse.

n = 8; % produce an 8x8 matrix, so small enough to display here
A1 = toeplitz([1;zeros(n-1,1)],[1 -1,zeros(1,n-2)]);
A1

A1 = 8×8

1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1

Or you could be glitzy, and not worry about the memory or even worry about efficiency.

A2 = ones(n);
A2 = triu(A2) - 2*triu(A2,1) + triu(A2,2);

A2

A2 = 8×8

1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1

Or you could use indexing, with sub2ind.

A3 = zeros(n);
A3(sub2ind([n,n],1:n,1:n)) = 1;
A3(sub2ind([n,n],1:n-1,2:n)) = -1;

A3

A3 = 8×8

1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1

Best, as I said, is to use spdiags, since this really is a bi-diagonal matrix. Create it using a tool designed to produce banded matrices. Matt showed you how.