MATLAB: Inverse of Matrix with condition…

matrix

I need to do the inverse of a lower triangular matrix for which I need to choose elements (a_ij) = (i+j)^2 when i>j. choose the length n=8. I know the inverse of lower triangular matrix but don't know how to choose the elements so that the above conditions are satisfied. Help me please…

Best Answer

   I=(1:8).';
   J=I.';
   A=tril( (I+J).^2 )

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MATLAB: Inverse eigen

Perhaps you could give an example of what you mean?

MATLAB: Matrix inverse using Cholesky decomposition

If you have the Cholesky factor, then just use back substitution (twice), applied to an identity matrix. Actually, it will be a forward substitution, then a back subs.

A = L*L'

where L is a lower triangular matrix. So, the problem is to compute the inverse, Ainv here:

L*L'*Ainv = eye(n,n)

Think of it as first solving the problem

L*u = eye(n,n)

L is lower triangular. The right hand side is known, the identity. Apply forward substitution to each column of the RHS.

Then solve a second problem.

L'*Ainv = u

Again, u is now completely known. L' is a upper triangular matrix. Use back substitution to solve for the square matrix Ainv.

Easy, peasy. If you got as far as the Cholesky factor, then this part will be easy for you.

When you are done, VERIFY YOUYR RESULT! Test that

A*Ainv = eye(n,n)

Don't worry if there is some floating point trash in the off diagonal elements of the product if they are on the order of 1e-16. That is to be expected.

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MATLAB: Inverse eigen

MATLAB: Matrix inverse using Cholesky decomposition

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