I need help with the following problem:
$A = I - uv^T$, where u and v are vectors in $R^n$. Find and expression for $\kappa_2(A)$ in terms if the inner products $r = (u, u)$, $s = (v, v)$ and c = $(u, v)$.
This is the way I approached the problem.
1) First finding the inverse of A. I found it to be $A^{-1} = I + \frac{uv^T}{1-v^Tu}$. This can be checked to be true by multiplying the inverse of A and A together.
2) Insert into the expression for the condition number $\kappa_2 = \|A\|_2\|A^{-1}\|_2$. Use that $\|A\|_2 = \sqrt{\rho(A^TA)}$, where $\rho$ is the spectral radius of $A^TA$.
This is where I am having problem and need help. How do I find an expression for the spectral radius of $A^TA$ and $(A^{-1})^TA^{-1}$ in terms of the inner products. The spectral radius of $A^TA$ is the same as the largest eigenvalue $|\lambda_{max}|$ of $A^TA$.
I do believe I am approaching this problem the correct way, however I am not able to find the eigenvalues of $A^TA$ and $(A^{-1})^TA^{-1}$. All help is appreciated.
Let's assume that $u$ and $v$ are not linearly dependent. The vector space of vectors that are orthogonal on $\mathrm{span}\{u,v\}$ has dimension $n-2$, and for each vector $w$ in this space we find $A^TAw=w.$ This means, $A^TA$ has eigenvalue $1$ with multiplicity (geometric as well as arithmetic) of at least $n-2.$
The other eigenvectors must be in $\mathrm{span}\{u,v\}.$ Let $w=au+bv.$ We want $A^TAw=\lambda w,$ which means $$(I-uv^T-vu^T+uv^Tvu^T)(au+bv)=\lambda (au+bv)$$ or $$ (a-ac+ars-bs+bcs)u+(b-ar-bc)v=\lambda(au+bv) $$ Since $u$ and $v$ are linearly independent, this can be rewritten as $$ \begin{pmatrix} 1-c+rs & s(c-1) \\ -r & 1-c \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \lambda \begin{pmatrix} a \\ b \end{pmatrix} $$ The eigenvalues of this $2\times 2$-Matrix are $$ \lambda_{1/2} = 1-c+\frac{rs}{2} \pm \sqrt{rs \left(1-c+\frac{rs}{4}\right)} $$ Now you have to set $\lambda_{\min} = \min\{1,\lambda_1,\lambda_2\}$ and $\lambda_{\max} = \max\{1,\lambda_1,\lambda_2\}$ and $\kappa_2(A) = \sqrt{\lambda_{\max}/\lambda_{\min}}.$
Finally, you also have to consider the case that $u$ and $v$ are linearly dependent or $u=0$ or $v=0.$