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Results tagged with matrix-equations
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user 36721
Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+CX+D=0$ or matrix differential equations (e.g. $\dot X(t)=AX(t)$. This tag is *not* meant for general systems of linear equations $Ax=b$.
2
votes
Accepted
Least square error problem ill conditioning
At least when $c:=x_i-y_i$ does not depend on $i$, any $n\times n$ submatrix of $A$ is of the form $M=(f_c(u_i-u_j)\colon i,j=1,\dots,n\}$, where $f_c(x):=f(x+c)$. So, $\det M>0$, since the function $ …
- 128k
4
votes
One observation of special type of square matrix exponentiation
(As is also now seen from your answer) I think in your question you actually wanted to impose the condition
$$\sum_{i=1}^j x_{ij} = 1\ \forall j \in \{2,\dots,n\} \tag{1}\label{1} $$
on the column sum …
- 128k
3
votes
One observation of special type of square matrix exponentiation
This is to complete the nice answer by tsnao by showing that $A_1^k\to0$ as $k\to\infty$.
To get that conclusion it is enough to assume that the $x_{ij}$'s are any complex numbers such that $$t:=\max_ …
- 128k
3
votes
One observation of special type of square matrix exponentiation
Based on the previous answers by tsnao and myself, one gets another, more elementary proof of your desired conclusion (and actually of a more general statement).
Indeed, by those previous answers,
$$L …
- 128k
2
votes
Accepted
Asymptotic behavior of a matrix equation and its eigenvalues
Of course, without assumptions on the behavior of the eigenvectors, your desired conclusion will not hold. E.g., for $t:=\lambda$, let
$$A(t):=\left(
\begin{array}{cc}
2+\cos t & \sin t \\
\sin t & …
- 128k
2
votes
Is it possible to simplify the coefficient matrix for large values of $x$?
Let $M:=M(x,y,z)$ be the $8\times8$ matrix in question. Let $m(x):=M(x,0,0)$.
We have
$$\det m(x)=
-256 e^{i x/2} x^2 \cos (2 x) \big((x^2+1)^2 \cos (2 x)-(x^2-1)^2\big).$$
So, $|\det m(x)|$ will be o …
- 128k