# Tag Info

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• 6,967

### One observation of special type of square matrix exponentiation

The answer is quite simple. First observe that $A$ is triangular, hence the spectrum is on the diagonal. From your assumptions, $1$ is a simple eigenvalue and the other eigenvalues belong to $[0,1)$. ...
• 51.8k
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### Is the set of real matrices with at least one real logarithm closed under multiplication?

This is already not true for $2$-by-$2$ matrices: Consider $$A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad \text{and}\quad B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}.$$...
• 107k
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### Sprinkling signs in unitary matrices

There are 5 inequivalent Hadamard matrices of order 16; if I understand correctly that's a counterexample.
• 19.5k
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Set $Y=K+MU$. You have $Y^2=U$, so $K+MY^2=K+MU=Y$, which gives an equation in $Y$ only: $$K-Y+MY^2 = 0$$ This is a widely studied equation; see for instance Higham and Kim, http://www.maths....
• 19.5k
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### Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices?

The answer is yes. Recall that the discriminant of a polynomial $x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ is a polynomial $\Delta(a_0, a_1, \ldots, a_{n-1})$ which vanishes if and only if the ...
• 152k
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• 5,837
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### Matrix equation with Hadamard product and its own inverse involved

Removing all unnecessary parameters, we come to the equation $\Omega^{-1}=2 W\odot \Omega + B$ where $B$ is positive definite. We need to find a solution in the cone $M_+$ of positive definite ...
• 60.6k

### Non-linear matrix equation

Comparing equation (1) with its transpose, we see that $AB X^T = X B^T A$. I would start by solving this linear equation.
• 53.9k

### The number of 0-1 normal matrices

For orders 1 to 9: 2, 8, 68, 1124, 36112, 2263268, 281249824, 70329901860, 35546752694048. I computed these numbers by finding representatives of the isomorphism classes of normal digraphs plus the ...
• 37.4k
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### Efficient algorithm for matrix equation $AXB + BXA = F$

For dense problems, the standard algorithm is a generalization of the Bartels--Stewart algorithm: see for instance https://doi.org/10.1016/S0024-3795(87)90314-4 and https://people.cs.umu.se/isak/recsy/...
• 19.5k
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• 19.5k
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### Determinants (and traces) of linear maps of matrices

We have $F(X)=\sum_i A_i X B_i = \sum_i (B_i^T \otimes A_i) vec(X)$ (see here), i.e. $F \sim \sum_i (B_i^T \otimes A_i)$. Because of some formulas here we have $tr(F)=\sum_i tr(A_i)tr(B_i)$. For the ...
Accepted

Actually, you have completely solved it yourself, just didn't dare to acknowledge it. In my notation, you have $(X\circ X^T)v(A)=(Y\circ Y^T)v(I)$ when $Y^2=XAX$. Similarly, $(Z\circ Z^T)v(I)=(Y\circ ... • 60.6k 5 votes Accepted ### Solving a vector of quadratic equations Shameless advertisement to a paper of mine: http://www.sciencedirect.com/science/article/pii/S0024379511004484 Quadratic vector equations, in Linear Algebra and its Applications, volume 438, 2013. ... • 19.5k 5 votes Accepted ### Trace of a nonlinear matrix equation If$A=I$, then it follows from the iteration rule that$\mathrm{tr}(X_{k})=1$for all natural number$k \ge 0$. If$\mathrm{tr}(X_{1})=\mathrm{tr}(X_{0})$, then the given iteration rule implies that:... • 6,015 5 votes Accepted ### Solution for$AX+XA^T+XBX=C $where$X$,$B$and$C$are symmetric That equation is called a (continuous-time) algebraic Riccati equation, and there is ample literature on when they are solvable; just look for this search term. For instance, the book Algebraic ... • 19.5k 5 votes Accepted ### Number of 5x5 matrix permutations without repetitions in rows or columns The answer to Question 1 is yes. What you have described is called a Latin square. Two Latin squares are isotopic if one can be obtained from the other by permuting rows, columns, and permuting the ... • 31.7k 5 votes ### A problem about determinant and matrix If there is a rational nonzero solution, there is an integer nonzero solution by multiplying up. At least one of the integers can be assumed odd by dividing out a common power of two. The determinant$...
• 37.4k

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