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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 …

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 …

So far, only the div-curl Lemma could be used when applying compensated compactness to systems of conservation laws. But it has two flaws. Because we only have divergences but no curls in general, w …

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 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 don't think that determinants is an old fashion topic. But the attitude towards them has changed along decades. Fifty years ago, one insisted on their practical calculation, by bare hands of course. …

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

P. D. Lax & B. Wendroff. On the stability of difference schemes, Communications on Pure and Applied Mathematics 15, pp 363–371 (1962).

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 …

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 …

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 …

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 …

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 …

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 …