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I've looked around for a study of the following matrix decomposition: $A = QUQ^T$ where $A$ is a complex matrix, $QQ^T = I$, and $U$ is an upper triangular matrix. It can be shown that not all matrice …

The Takagi decomposition provides a canonical form for a complex symmetric matrix $S$ under $U \mapsto USU^T$ where $UU^* = I$. Question: Is there an anti-Takagi decomposition? I.e. Is there a canonic …

I've come up with the following piece of Python code (using the library Sympy): def double_diagonalize(m1, m2): V, _ = (m1.T * m2).diagonalize() U, _ = (m1 * m2.T).diagonalize() return U, …

It's well known that the following set of real square matrices is dense: those matrices $M$ for which there exists an invertible matrix $P$ such that $P M P^{-1}$ is diagonal. My question is can this …

Chapter XI Theorem 3 from here implicitly states that an invertible complex symmetric matrix always has a complex symmetric square root. It's clear that a square root exists, by appealing to the Jorda …

Maybe this question is too elementary or too vague, but there might be something interesting here: A $2 \times 2$ Jordan matrix is of the form $\begin{pmatrix} \lambda & 1 \\ 0 & \lambda\end{pmatrix} …

Every real square matrix $M$ has a QR decomposition $M = QR$ where $Q^{-1}=Q^T$ and $R$ is an upper triangular matrix with non-negative reals on the diagonal. Call the function $f(QR)=RQ$ the Francis …

Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex numbe …

Let $A$ be an arbitrary real square matrix. Does there exist an $\mathcal O(n^3)$ algorithm for deciding whether there exists a permutation matrix $P$, lower unit triangular matrix $L$, upper unit tri …

Given an arbitrary complex matrix $M$ and real, diagonal but possibly indefinite matrix $\Delta$, the problem is to solve the following system of equations: $$\begin{aligned} M^*\Delta M &= LD^2L^*\\ …

It's well known that LU decomposition is only numerically stable if it's combined with row and/or column pivoting. It makes me wonder if there are other matrix decompositions that can profitably be co …

I don't know any abstract algebraists personally, which is why I'm asking this question here. Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated …

Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is equal to $ …

Given an unordered pair of complex numbers $\{w,z\}$, we can associate to it the complex matrix $$\frac 1 2\left[\begin{matrix}w + z + \frac{\left(w - z\right)^{2} + \left|{w - z}\right|^{2}}{2 \left| …

Given a symmetric Hessenberg matrix $A = \left[\begin{matrix}\ddots& \vdots & \vdots\\\dotsb & a & b\\\dotsb& b & c\end{matrix}\right]$, the Wilkinson shift $\mu$ employed in some eigenvalue solvers i …