Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $A$ is a real $n\times n$ normal matrix, and consider the matrix $$B=(I_n-\theta A)^{-1}(A+A^T)(I_n-\theta A^T)^{-1},$$ where $\theta$ is a real scalar such that $(I_n-\theta A)$ is invertible.

Since $B$ is symmetric, I can write $$B=H\Lambda H^T,$$ where $\Lambda$ is a diagonal matrix containing the eigenvalues of $B$ and $H$ is an orthogonal matrix containing real eigenvectors of $B$.

My question is: how is $H$ related to the eigenvectors of $A$, or of ($A+A^T$)?

(Letting $A=UDU^{*}$, where $U$ is unitary and $D$ is the diagonal matrix containing the eigenvalues $d_i$ of $A$, we see that the eigenvalues of $B$ are $|1-\theta d_i|^{-1}\mathrm{Re}(d_i)$. It's the eigenvectors I'm not sure about. )

share|cite|improve this question
There's something wrong with your question. Let $A=I_n$, $A$ is a normal matrix, but $I_n-A$ isn't invertible. – Git Gud Feb 6 '13 at 18:40
ok thanks, edited – mark Feb 6 '13 at 18:54

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.