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You cannot do much, for the following reason. Take any real symmetric matrix $M$. It is a consequence of the Toepliz-Hausdorff Theorem about the Numerical Range that $M$ is unitarily similar to a (sym …

Weyl's inequalities are not the full story. The characterization of the possible spectra of $A+B$, given the spectra of $A$ and $B$ is the object of A. Horn's conjecture. This is now a theorem, after …

As far as DFT and FFT is considered as a part of Numerical Analysis, finite abelian groups apply, especially ${\mathbb Z}/2^n{\mathbb Z}$.

Relaxation method is a part of the theory of iterative methods for the calculation of approximate solutions to linear systems. It depends on a real or complex parameter $\omega$, the relaxation parame …

The QR algorithm gives quite rapidly a good approximation of the eigenvalues of a real symmetric (or complexHermitian). But overall, it gives first the smallest eigenvalues. The reason is that the rat …

The Newton's method has driven a lot of attention during the past decades in the community of complex analysis, because of its nice behaviour as a dynamical system. From the point of view of the appro …

If the solution of the BVP is non-unique, it is likely that the solution of the discretization is non unique as well. I should bet on the following. If a solution of the BVP has the (generic) property …

Since your approximate solution is piecewise linear, it is $H^1$ but not $H^2$. Therefore your calculation is impossible. You can do to things to overcome the difficulty: Either use higher-order ele …

This is an interesting question, to which the answer is positive. Here is the proof. Of course, for the method to make sense, we must assume that the diagonal $D>0$. The notations below are borrowed …

I give a course on matrix analysis, and I like to mention the following thing. It is equivalent to finding the roots of polynomials, or to finding the eigenvalues of matrices, because to a polynomial …

This completes Polini's answer, which is perfectly right, but a bit 'elliptic'. The QR algorithm is usually employed together with a preparation step: one puts the matrix $A$ in a unitarilly similar H …

The ambiguity in your question is the word 'rapidly'. If you want to have an information on the eigenvalues of $A+B$, without any extra information besides those given in the question, then this is t …

The set of matrices $A$ is a cone, smooth away from the origin. With the Frobenius norm $\sqrt{{\rm Tr}(M^TM)}$, you can use differential calculus. The minimum is achieved at some $A$. If $A\ne0_n$, t …

The answer is yes, because your function (let me call it $f$) is the maximum of convex functions. As such, it is convex. The formula : $$f(X)=\max\left(0,\max_{1\le r\le n}\sum_{j=1}^r\lambda_j(X)\rig …

There is an interesting though partial answer to your question: Under your assumptions, it is not possible that the signature of $A$ be $(n-1,1)$ (exactly one negative eigenvalue). Proof: With s …