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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 $Ax=b$.
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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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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4
votes
Accepted
Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric
Let $spectrum(A)=(\lambda_i)$, $\phi:X\mapsto AX+XA$ and $K$ be the condition number of $A$ for $||.||_2$.
We assume that $A$ is positive definite.
Then $spectrum(\phi)=(\lambda_i+\lambda_j)_{i,j}\sub …
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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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2
votes
Matrix equation
A necessary condition. Note that $D=diag(\dfrac{q_i}{a_j^TCa_j})$ is symmetric $>0$.
$C^{-1}=ADA^T$ is symmetric $>0$; then $C$ is symmetric $>0$. One has $A\sqrt{D}\sqrt{D}A^T\sqrt{C}\sqrt{C}=I_k$, …
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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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2
votes
Accepted
Solvability of $A X B=C$ with $X=X^\mathrm{T}$
Here $U^+$ denotes the Moore-Penrose inverse of $U$.
Assume that $A,B,C$ are generic; there exists a solution $U$ of $AU_{n,k}=C$ only if $AA^+C=C$.
This -generically- never happens unless $AA^+=I_m$, …
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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
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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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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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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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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1
vote
Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$
Two permutations are conjugated in $S_n$ iff they have same spectrum (as matrices).
Since $A,B_1,B_2$ are symmetric, they are a product of $k$ (the same $k$ for all) disjoint transpositions.
Let $A,B_ …
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0
votes
Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?
As Michael wrote, we may assume $c=1$.
Your equation can be written $diag(\phi(X))=[1]$ where $\phi$ is the linear function $\sum_{k=0}^{\infty}A^k\otimes A^k=\sum_{k=0}^{\infty}(A\otimes A)^k$ (Her …
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