Symmetric linear least-squares solution

I assume that $A$ is onto, so that $H:=A^TA$ is positive definite. Minimizing $\|AX-Y\|_F^2$ in Frobenius norm (the least square) among symmetric matrices $X$ yields the optimality condition that $$\langle AS,AX-Y\rangle=0$$ for every symmetric $S$. This amounts to saying that $A^T(AX-Y)$ is skew-symmetric. In other words, $X$ is the solution of the Lyapunov equation $$HX+XH=A^TY+Y^TA=:K.$$ The explicit formula is $$X=\int_0^\infty e^{-tH}Ke^{-tH}dt.$$

Tags:

Least Squares

Quadratic Programming

Matrices

Matrix Equations

Related

Reduction to Lie algebra version of fundamental lemma? Every possible number of partitions by restricting parts? Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$ Is every metric uniformly close to a metric with negative scalar curvature? Strange formulas that gave rise to Koszul duality Looking for a simple one-dimensional noetherian domain whose regular locus is not open Which right square pyramids are scissors congruent to a cube? Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function? Is the equational theory of groups axiomatized by the associative law? $|(A+B)(X)|=o(X)$ if $|A(X)|=O(X^{1/2})$ and $|B(X)|=O(X^{1/2})$? Norms as Points in $C(X)$ Can different knots have the same numbers of quandle colorings for all quandles?

Recent Posts