# Federico Poloni

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 Name Federico Poloni Member for 3 years Seen 40 mins ago Website Location Pisa, Italy Age 30
Numerical linear algebra. Matrix equations + control.
 16h awarded ● Nice Answer 1d comment how to solve this multivariate quadratic equation?Can you add a link to the discussion on math.stackexchange? 2d comment Eigenvalues of Symmetric Tridiagonal MatricesAnother argument is: it's easy to reduce any symmetric matrix to tridiagonal with similarity transforms. So if this problem were easy to solve, all symmetric eigenproblems would be. 2d comment Solve $(A+B)x=y$ given Cholesky decomposition of A and BIn general, no. One can make low-rank updates of factorization easily, but for anything different the usual answer is "you have to compute everything from scratch". Maybe you can use CG with $A$ (or $B$) as a preconditioner, but that will be numerically convenient only in some special cases. 2d comment fixedpoint or fixed point or fixed-pointI am not familiar with that definition --- I call pre-fixed point a point $x$ such that $f(x)\neq x$, but $f^k(x)=f^{k+1}(x)$ for some $k\geq 1$. 2d answered Why don’t more mathematicians improve Wikipedia articles? 2d comment fixedpoint or fixed point or fixed-point"prefixed" here most likely comes from "pre-fixed", not "prefix+ed". I guess this shows once more the importance of using hyphens. :) May22 comment What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ?Related question: mathoverflow.net/questions/63027/…. I'd start from the book suggested in the answer, Horn and Johnson's Topics in Matrix Analysis (not to be confused with Matrix Analysis by the same authors). May20 comment What does “Vertex Solution” mean?A bit more context would help. May17 comment How to maximize the determinant of a matrix of the form VDV^H Not sure about that: $V^HV$ is a rank-$M$ $2M\times 2M$ matrix, so its determinant would be zero. May15 comment spectral radius monotonicityYou are right! One needs an additional hypothesis (irreducibility) to get strict inequalities in P-F. May15 revised spectral radius monotonicityadded 15 characters in body May14 revised Can an accumulation point be an eigenvalue?deleted 20 characters in body May13 comment Eigenvalues of random Hamiltonian matricesI must be missing something then. Doesn't conjecture (1) (proved in a paper) say that they should be uniformly distributed in the limit? May13 comment Eigenvalues of random Hamiltonian matricesIt is curious that there seems to be an area with less eigenvalues around the imaginary axis (for Hamiltonian) or the real axis (for Ginibre). Is it an artifact of the numerical methods? I can imagine them having trouble in identifying properly purely imaginary/real eigenvalues. May6 comment Spectral Properties of $A(I-A)^{-1}$T is triangular. en.wikipedia.org/wiki/Schur_form May6 accepted spectral radius monotonicity May5 answered spectral radius monotonicity May1 comment Problems where Conjugate gradient works much better than GMRESFrom the look of it, yes, but there is probably a completely theoretical question in matrix approximation theory hidden behind this one. Apr29 comment On solution of a discrete-time equationFor a general linear matrix equation with more than two terms, I am not aware of any method faster than the $O(n^6)$ of the naive "vectorize everything" algorithm. (caveat: I know almost nothing of LMIs, so there might be a method in that area that I do not know about). Maybe you'll have more luck posting this as a separate question, either here or on scicomp.stackexchange.com. Apr29 comment On solution of a discrete-time equationit more stable, and ultimately obtain a discrete-time version of the ADI method for Lyapunov equations. In short, with some algebraic manipulations everything that works in the continuous-time case rates to work in the discrete-time case as well. Apr29 comment On solution of a discrete-time equationYep. dlyap will be fine for a small-scale problem; it should use a variant of the same Bartels-Stewart method that is used for continous-time lyapunov eqs; essentially, take a Schur form of $F$ and solve directly via back-substitution for each entry of $X$ "in the right order". For large-scale problems, you can truncate the series $X=\sum_{i=0}^{\infty} F^{i}CF^{Ti}$, or obtain the partial sum truncated at the term $2^{k}$ directly from the one truncated at $2^{k-1}$ with some manipulations (Smith methods). You can apply some Möbius transforms to $F$ without changing the solution to make... Apr28 comment On solution of a discrete-time equationThere are numerical methods dealing directly with the equation $X-FXF^T=C$, known as discrete-time Lyapunov equation, or Stein equation. There is no need to invert $F$ and convert it to a Sylvester; sometimes algorithms even go the other way round and convert a continuous-time Lyapunov equation to a discrete-time one to solve it. Apr25 comment naming for the map $T = x \mapsto a x b$Since you are back here: thought that the name "discrete-time Sylvester operator" makes a lot of sense for the map $S:X\to X-AXB$, since a discrete-time Sylvester equation can be written $S(x)=C$, analogously with other similarly-named operator. The name with commas returns a whopping two Google results, but I think that a person working in matrix equation would understand immediately what you're talking about if you use it. Apr22 comment The first eigenvalue of a branching process matrixYou are right, sorry for forgetting this part! Apr22 answered The first eigenvalue of a branching process matrix Apr22 comment The first eigenvalue of a branching process matrixThe second "smaller" should be "larger". Apr21 comment On solution of a recursion with rectangular matricesWhat is the source of this problem? Looks a lot like a discrete-time difference Riccati equation. This, and what Robert Israel said. Apr18 comment Products of matrices of a certain formThey also happen to be semiseparable. Apr18 comment Control a linear system to the kernal space of the output matrix.Maybe it would help if you added $(t)$ and quantifiers such as $\forall t \in \mathbb{R}$ in the right places... Apr17 comment Control a linear system to the kernal space of the output matrix.If $Cx=0 \Rightarrow x=0$, then the two tasks are equivalent, isn't it? Is there maybe a typo in your question? Apr16 accepted Efficient computation of Markov chain transition probability matrix Apr15 answered Efficient computation of Markov chain transition probability matrix Apr14 revised Old books still usedpublication date edited in, mathscinet cites Apr8 comment Global solution for spectral clusteringPlease don't cross-post to two different sites before waiting at least a few days without answers. Your question is only 3 hours old, and you already got a seemingly good answer on math.se. Apr8 comment Numerical evaluation of a triple integral can be made ?What is your question exactly? You can always write that expression; of course it will not be equal to the exact integral, but it will converge to it (slowly) under mild hypotheses. Apr2 comment Strassen’s algorithmIf you remove it, and work in finite precision arithmetic, then you can replace an arithmetic operation with a bazillion if's and assignments. But I guess that it can be removed, if one restricts to algorithms that do not "cheat" using real number representations. Apr2 comment Strassen’s algorithmThat is the same algorithm that I had in mind. Apr1 comment Strassen’s algorithm@Dima Pasechnik not necessarily. For instance, you can easily compute the square of a real 2x2 matrix with only 5 multiplications, but this is an algorithm that exploits commutativity, and thus cannot be recast in bilinear "tensor language". Apr1 answered Strassen’s algorithm Mar31 comment Does this matrix shape have a name?@AaronMeyerowitz not that $J$ is a standard notation for the all-ones matrix... Mar15 revised Multiple Number Partitioning / “Multiprocessor Scheduling”typo in tag Mar6 accepted Nth root of a matrix as an analytic function? Feb24 answered Nth root of a matrix as an analytic function? Feb20 accepted Sum of elements of inverse matrix Feb19 revised Fiction books about mathematicians?expanded, incorporated suggestions Feb19 answered Fiction books about mathematicians? Feb18 answered Sum of elements of inverse matrix Feb1 answered Compute roots of sum_i c_i/(a_i + b_i x)^p Jan24 asked Kronecker-structured matrix kernel