Efficient Solution for (A+pI)x=b: Fast Matrix Inversion

[blindconsole]

Hello,

I'm trying to find a fast way to solve the matrix equation (A+pI)x=b, where A is a large matrix, I is the identity matrix, and p is a parameter whose value needs to be swept. Obviously I could just use mldivide or matrix inversion for every value of p, but this seems inefficient. Does anyone know of a better way? Thanks!

 

[Dickfore]

Diagonalize the matrix A:
<br /> A \, U = U \, \Lambda<br />
where \Lambda is a diagonal matrix whose diagonal entries are the eigenvalues \lambda_{i} of A and U is a vector whose columns are the eigenvectors of A. Then, we have:
<br /> f(A) = U \, f(\Lambda) \, U^{-1}<br />
where by f(\Lambda) we mean a diagonal matrix whose diagonal entries are f(\lambda_{i}).

Now, notice that if you have A, diagonlaized, then:
<br /> (A + p I) \, U = A \, U + p I \, U = U \, \Lambda + p U \, I = U \, (\Lambda + p I)<br />
where we have used the fact that I U = U I = U. Here, \Lambda + p I is again a diagonal matrix whose eigenvalues are \lambda_{i} + p. Then, we will have:
<br /> (A + p I)^{-1} = U \, \frac{1}{\Lambda + p I} \, U^{-1}<br />

 

[Stephen Tashi]

I don't know if the answers on the following web page are correct, but they look interesting:http://math.stackexchange.com/questions/17776/inverse-of-the-sum-of-matrices