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Results tagged with matrix-equations
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user 9091
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 $Ax=b$.
1
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Can the nonlinear matrix equation $Bx=p+\text{sgn}(x)$ have an explicit solution?
I assume that $\text{sgn}(a)=1$ if $a>0$, $\text{sgn}(a)=-1$ if $a<0$, $\text{sgn}(0)=0$ and $\text{sgn}([x_1,\cdots,x_n]^T)=[\text{sgn}(x_1),\cdots,\text{sgn}(x_n)]^T$.
In the sequel I assume that w …
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2
votes
Solving a quadratic matrix equation
I read the reference of Igor, but I have nothing shot about our equation !
There is a case that seems less difficult. If $A,B$ are symmetric complex, then the problem is essentially equivalent to so …
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4
votes
Accepted
Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution
We can solve quickly this problem using the basis Grobner theory.
Put $X=[x_{i,j}],Y=[y_{i,j}]$.
We consider -over a field of characteristic $2$- the algebraic system in the $128$ unknowns $x_{i,j},y_ …
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0
votes
How to obtain matrix from summation inverse equation
If $n\geq 2$, generically, you are unlikely to encounter such an equation with only one solution.
Let $d$ be the dimension of our symmetric matrices.
$\bullet$ Consider the case $n=2$. Then the algebr …
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6
votes
How to solve this quadratic matrix equation?
In fact, your problem reduces to a Riccati equation in dimenion $n-1$ that is easy to solve. Here, if I correctly understood your problem, $A=I_n+aa^T, C=(cc^T)(cc^T)=uu^T$ (since $rank(C)=1$), where …
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-1
votes
Solving a quadratic equation for an hermitian matrix
This is a very partial answer. Assume that $S$ is invertible ; then $S$ and $I$ are congruent and there is $K$ s.t. $K^TK=S$. Here a solution $H$ must be a hermitian matrix ; then $HH^T=\overline{S}$. …
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1
vote
Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D
Remark 1. The eigenvalues of $A\otimes B +B\otimes A +cI$ are not the $\alpha_i\beta_j+\beta_i\alpha_j+c$ except when (for example) $AB=BA$.
Remark 2. The complexity of your problem is not in $O(p^6) …
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3
votes
Non-linear matrix equation
In this post, I assume that $A$ is symmetric $>0$ (in particular invertible) and $B$ is invertible.
$\textbf{Proposition}$. i) Equation (1) generically admits $2^n$ real solutions $X$.
ii) There are …
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1
vote
Accepted
Conditions for a certain matrix equation to have a full rank solution
We ignore the supplementary condition $\sum_lA_lA_l^*=I$. Adding conditions 24 hours after the first post is making fun of the world.
We assume that $n,m,t$ are fixed positive inytegers s.t.$m,t\geq …
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1
vote
System of quadratic equations
When the $(B_i)$ are in general position, there are always $2^s$ complex solutions but we don't have control over the number of real solutions.
Let $n=2^{s-1}$. In generic cases, the system can be dec …
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0
votes
Accepted
Product of matrices equal identity
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\rank{rank}$Here is how we can construct solutions in $P$.
Necessarily, $r\leq d$, $\rank(P)=r$, $\rank(S)=r_1\geq r$.
There is $Q$ invertible s.t. …
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0
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Solving linear matrix equation
I feel like you are drowning in a glass of water.
Putting $AC=U\in M_2$ and $BC=V\in M_2$, we obtain
(*) $UXU^T-VXV^T=L$ in $M_2$.
If $X$ is a symmetric solution of (*), then $L$ too.
For generic $U,V …
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2
votes
Diagonal Lyapunov equation with rank 1
It seems hopeless to me. The considered equation
$A^TPA-P=bb^T$ can be rewritten $\phi(P)=bb^T$, where $\phi=(A^T\otimes A^T-I_{n^2})$ -if we stack the matrices row by row into vectors-.
If $spect …
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1
vote
Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters
There are $P\in U(n),Q\in U(p)$ s.t. $A=P^*\Delta P,Q^*BQ=D$ where $\Delta,D$ are diagonal real; then
$Q^*X^*P^*\Delta PXQ=D$, that is
$(1)$ $U^*\Delta U=D$, where $PXQ=U=[u_{i,j}]\in M_{n,p}$ or …
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0
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Determine unknown matrix function of particular form from known points
We have a black box $f:X\mapsto B^{-1}(X-A)(DB^{-1}(X-A)+C)^{-1}$.
We are looking for approximations of $A,B,C,D\in M_n$, where $A,B,C$ are invertible.
Note that if $(A,B,C,D)$ is a solution, then $(A …
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