Updating eigen decomposition for a matrix after some row changes

Question

Let us say we have a matrix $A$ which has eigen decomposition

$$A=UDU^{-1}$$

If some of the rows of A are changed by multiplying a constant positive value, is there a simple way to update the eigen decomposition from existing $D$ and $U$?

Many thanks.

score 2 · Accepted Answer · answered Oct 20 '13 at 12:50

2

There is none that I know of. What you ask about is $XA$, where $X$ is a diagonal matrix.

Note that by premultiplying $A$ with some diagonal matrix $D$, you generally lose Hermitianity/similarity/normality/... and any other nice structure that $A$ may have had.

So, while $A$ may have only real eigenvalues (for $A$ Hermitian/symmetric) or on a unit circle (for $A$ unitary/orthogonal), $XA$ can have random complex eigenvalues.

Further, when you lose normality ($A^*A = AA^*$, but $(XA)^*(XA) \ne (XA)(XA)^*$), you also lose diagonality of $D$.

answered Oct 20 '13 at 12:50

Vedran Šego

11,372

Thanks Vedran. I have updated my question. – webappl Oct 21 '13 at 20:22
This is a completely new question, and it should be opened as such, possibly with a link to this one, but leaving this one as it was. This way, my answer is meaningless. – Vedran Šego Oct 21 '13 at 21:25
I moved my new question to a new post. link – webappl Oct 21 '13 at 21:42
Thank you. I appreciate it. – Vedran Šego Oct 21 '13 at 21:44

Updating eigen decomposition for a matrix after some row changes

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