Let $T ∈ R^{n×n}$ be symmetric tridiagonal, let $z ∈ R^n$ , and let $A = T + zz^T.$ Show how to use the symmetric Lanczos algorithm to produce an orthogonal $V ∈ R^{n×n}$ such that $V^TAV = \hat{T}$ where $\hat{T}$ is tridiagonal. Your procedure should require $O(n^2)$ operations. What is the ideal choice for $v_1 = V e_1$ for this procedure?
In the above question, what should be the ideal choice for $v_1$? Isn't it supposed to be a random unit vector?
And what is the significance of $A = T + zz^T$? Can't it be applied to any hermitian matrix A?