All Questions
Tagged with matrices oc.optimization-and-control
107 questions
1
vote
0
answers
63
views
The rank of a matrix expression
I'm studying discrete-time LTI systems and state estimators for them. Recently, I studied this paper. I am facing a matrix rank calculation problem and having trouble solving it. I will provide more ...
- 4,484
0
votes
0
answers
53
views
Relations between the optimal solutions of two related SDPs
In system theory, we often encounter Semi-Definite Programs (SDPs) with Linear Matrix Inequality (LMI) constraints, such as those presented in this paper. I have introduced new variables based on the ...
- 4,484
1
vote
1
answer
67
views
Low rank matrices which preserve maximizers of quadratic form
Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form,
\begin{align}
x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y)
\end{align}
Does there exist an $M$ such ...
- 297
2
votes
0
answers
72
views
Gradient descent over the set of complex symmetric matrices
In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...
- 21
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
1
vote
1
answer
519
views
Optimal transport between two matrices
I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
- 243
1
vote
2
answers
69
views
Monotonicity of kernel matrices with respect to hyperparameters
Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
- 801
1
vote
0
answers
49
views
Bounds on Eigenvalues After Skew-Symmetric Perturbation
Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum:
$$\mathbf{A} = \...
- 11
1
vote
3
answers
345
views
How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?
Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix:
$$
\min_{s\in\...
- 723
3
votes
2
answers
246
views
$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$
(The construction of matrix $\mathbf{A}$ is not difficult to be understood. You can first jump to A Toy Example to take a glance. Any idea or suggestion would be appealing for me.)
The Original ...
- 133
5
votes
2
answers
343
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
- 51
0
votes
0
answers
159
views
Double summation of matrices as constraints in convex optimization in CVX
I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53:
\begin{align}
\text{minimize} &\qquad s\\
\text{subject to} & \...
4
votes
1
answer
228
views
A question on eigenvalue of parametric matrix
Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
- 1,216
3
votes
1
answer
255
views
Minimizing quadratic objective under orthogonality constraints
The following problem is motivated from Generalized Procrustes Analysis. I am wondering if it is possible to obtain a closed form minimizer (which may involve SVD or some other decomposition of a ...
- 161
8
votes
2
answers
380
views
Projecting onto space of matrices with spectral radius less than one
Consider the space
$$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$
where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...
- 123
1
vote
1
answer
1k
views
Prove that absolute value of eigenvalue is smaller than 1 [closed]
I want to prove that the absolute value of the eigenvalues of a matrix A are smaller than 1 for $$A=\left(\begin{array}{cc}
0 & -H_{11}^{-1} H_{12} \\
-H_{22}^{-1} H_{21} & 0
\end{array}\right)...
0
votes
0
answers
46
views
Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
0
votes
0
answers
193
views
Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
1
vote
0
answers
78
views
Is there an efficient way to do semidefinite programming with a Lyapunov equation constraint?
I am trying to numerically solve semidefinite programs of the form
$$\begin{array}{ll} \underset{X,Y}{\text{minimize}} & \operatorname{tr}(AX)\\ \text{subject to} & BY + YB = X\\ & X, Y \...
- 31
0
votes
2
answers
1k
views
Matrix norm minimization and matrix inner product
One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, Convex Optimization, p. 170).
Consider:
\begin{equation}\label{eq:Lasse}
\begin{aligned}
&\min_{\mathbf{x}}
& &...
- 345
0
votes
0
answers
113
views
Reference for matrix Lyapunov function / matrix dynamic system / stability
We usually consider $\dot{x} = f(x)$, where $x$ is a vector.
Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...
- 101
2
votes
1
answer
139
views
linear programming with $n$ choose $r$ variables
Given parameters $r < n$, define $m = {n \choose r}$ and let $A$ be the $n\times m$ matrix whose columns are all the vectors with $r$ $1$'s and $n-r$ $0$'s. Let $b$ be a positive $n$-vector. Is ...
- 51
1
vote
2
answers
87
views
Stability of linear controller in the presence of a lag
Consider the following equation:
$\ddot{x} = -a x - b \dot{x}$
which we interpret as saying that we are trying to control $x$ by setting $\ddot{x}$.
We can rewrite this with $X = \begin{bmatrix}
...
- 427
1
vote
0
answers
514
views
Minimizing the Frobenius norm of a quadratic matrix expression
Given matrices $R \in \mathbb R^{m \times n}$ and $Y \in \mathbb R^{p \times n}$, where $R$ is full rank, how can I solve the following optimization problems?
$$\min_{X \in \mathbb R^{p \times m}} \| ...
- 11
0
votes
1
answer
194
views
Optimization problem involving matrix
I am struggling to solve an optimization problem of the following form:
$$\begin{array}{ll} \underset{A}{\text{maximize}} & \log \det (A) \\ \text{subject to} & a^T A^{-1} a \le b\end{array}$$
...
- 13
5
votes
1
answer
261
views
Bounds on number of "non-metric" entries in matrices
Question:
what upper bounds are known on the number of non-metric entries of finite dimensional square matrices $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with strictly positive off-diagonal elements $...
- 13.2k
4
votes
1
answer
254
views
Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices
For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices:
$$\Pi(A)=\mathrm{argmin}_{M\...
- 41
0
votes
0
answers
95
views
How to maximum L1 norm problem?
I have met a problem these days.
\begin{equation}
\underset{\omega}{\max} \quad \Vert \text{diag}(\mathbf{h}^H)\mathbf{G}^H\mathbf{\omega}\Vert_1 \\
s.t.\quad\mathbf{\omega}^H\mathbf{G}\mathbf{G}^H\...
- 171
2
votes
1
answer
536
views
Minimize matrix norm over the unitary matrices
Suppose $C_1$ and $C_2$ are some fixed $n \times n$ matrices. Define the norm $\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$. What is $\min_U \|C_1 U C_2 \|$? Here $U$ ranges over the $n \times n$ unitary ...
- 1,703
1
vote
0
answers
152
views
solving a non-linear Matrix equation
I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...
- 377
5
votes
1
answer
335
views
Projecting a symmetric matrix onto the space of linear operators with a particular eigenvalue
Specifically, I am interested in the case where one eigenvalue is exactly $0$. Given an $n \times n$ symmetric matrix, I would like to find the closest $n\times n$ symmetric matrix that has one ...
- 245
2
votes
1
answer
497
views
Does the Perron vector maximize $x^TAx$ in the simplex?
Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^...
- 1,421
1
vote
1
answer
877
views
Minimization problem involving the inverse of an affine matrix function
I want to minimize $v^T (A+I+UQU^*)^{-1} v$, subject to $Q$ and $A$ being positive semi-definite and ${\rm trace}(Q)<1$. Here, $v$ is a given vector with unit norm, that is, $\|v\|_2=1$.
- 377
3
votes
0
answers
111
views
Approximate inverse of large sparse matrix
Given a large sparse matrix $M$, how to determine the existence of a good preconditioner? In other words, does there exist a sparse matrix $X$ such that $X M$ is close to the identity with respect to ...
- 31
1
vote
2
answers
143
views
Controllability Gramian asymptotics for small times
Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
- 724
0
votes
1
answer
135
views
Solving Problem: LMIs and block matrices
I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
- 103
4
votes
0
answers
249
views
Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
- 2,712
12
votes
2
answers
799
views
A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
- 2,712
2
votes
0
answers
248
views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...
- 2,712
3
votes
0
answers
243
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
9
votes
2
answers
684
views
A trace-constrained maximization problem in the cone of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
- 2,712
1
vote
0
answers
163
views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
- 11
1
vote
2
answers
307
views
How to solve this system of Matrix equations? (Coupled riccati equations?)
I am trying to solve for K in the following problem:
$ 3I = A_1 + A_2 + A_3$
$ A_1 K A_1 = K_1 $
$ A_2 K A_2 = K_2 $
$ A_3 K A_3 = K_3 $
Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...
- 13
8
votes
5
answers
481
views
Nearest matrix orthogonally similar to a given matrix
Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...
- 13.9k
3
votes
2
answers
2k
views
Proving that a matrix is positive semidefinite
Let matrices $A, B$ be positive semidefinite. Can we prove that $A(I+BA)^{-1}$ is positive semidefinite?
- 41
2
votes
1
answer
134
views
Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters
Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$.
Find matrix $X$ such that $X^*AX=B.$
Solution in the case of positive definite $A$ and $...
- 21
4
votes
1
answer
289
views
A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
- 81
1
vote
1
answer
313
views
Nonlinear low-rank approximation - corrected
I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed.
In my research of linear ...
- 620
1
vote
1
answer
124
views
Possible analytical way to solve or approximate a specific optimization problem's solution
In my research on linear algebra and optimization, I have come across the following problem repeatedly:
Given constant matrices $C\in\mathbb{R}^{k \times k}$ and $X\in\mathbb{R}^{n \times n}$, $$\...
- 620
0
votes
1
answer
134
views
Modification of a known optimization problem
In my research of linear algebra and optimization, I wish to modify the following well-known problem:
$ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector ...
- 620