Questions tagged [hurwitz-matrices]

A square matrix $A$ is a Hurwitz matrix if all eigenvalues of $A$ have strictly negative real parts.

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Are these two block matrices Hurwitz?

Define$$H_1=\left( \begin{matrix} -A & B \\ -{{B}^{T}} & 0 \\ \end{matrix} \right),$$ where $A\in {{\mathbb{R}}^{n\times n}}$ is a symmetric positive definite matrix. $B\in {{\mathbb{R}...
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Hurwitz stability status of two matrices

I have a complex symmetric matrix (it is not Hermitian), i.e. $\textbf{A}\in\mathbb{C}^{n\times n}$. Can you prove that $\textbf{A}$ and $\textbf{B}=\textbf{A}+\textbf{A}^*$ have similar Hurwitz ...
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About linear subspace of Hurwitz Matrix Manifold

In this question I want to investigate the linear subspace of the Hurwitz matrix family. That is to say, suppose $M_H = \{A \in R^{n \times n}: \operatorname{Re} \lambda_i(A) \leq 0, \forall i \leq n\}...
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Exponential of irreducible, aperiodic Hurwitz/Metzler matrix is irreducible or has no zero entries?

I am working with a Hurwitz-stable, Metzler matrix $A$ with nonpositive diagonal ($A_{ii}\leq0$ for all $i$) and nonegative off diagonal ($A_{ij}\geq0$ for all $i\neq j$). I want its exponential to be ...
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Prove that anti-triangular block matrix with a Hurwitz block is also Hurwitz

The closed-loop matrix of a dynamical system is an anti-triangular block matrix of the form $$ A_\mathrm{cl} = \begin{bmatrix} A & BS \\ -\epsilon \left( G_u S \right)^\top {M} C & 0 \end{...
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Preserving Hurwitz stability of block matrices with same eigenvalues of each block

Let us assume that $$A=-\text{diag}(d_1,\ldots,d_m) +Q \text{ is Hurwitz},$$ where $d_i>0, \forall i\in \{1,\ldots,m\}$ and $Q\in \mathbb R^{m \times m}$ is a possibly full matrix. Can we deduce ...
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Eigenvalues of symmetric part of product of matrices

Consider a real matrix $B$ defined as $$ B := X A + A^T X $$ where $X$ is a symmetric positive definite matrix and $A$ has eigenvalues with positive real parts. How can I prove that eigenvalues of $B$ ...
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How to determine whether all real parts of the eigenvalues of a complex matrix are negative?

It is well-known with respect to Routh-Hurwitz Criterion that for an arbitrary matrix $A$ with real coefficients, one can derive a series of analytic expressions with these real coefficients, so as to ...
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Show that every matrix with real eigenvalue less than $0$ is similar to dissipative matrix

Suppose $A \in \mathbb{R}^{n \times n}$. $A$ is dissipative if $A+A^T \prec 0$. How do I show that $A$ is similar to a dissipative matrix? I started by showing that the eigenvalue of $A$ is real and ...
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Estimates for eigenvalues of leading principal submatrices of Hurwitz matrices

In mathematical control theory a Hurwitz matrix or stability matrix $A$ for a asymtotically stable differential equation $ \dot{x} = A x $ has strictly negative real parts of eigenvalues $ \Re(\...
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A problem about positive definite matrices

Given $A\in\mathbb{R}^{n\times n}$, show that all eigenvalues of A has negative real part if and only if for each positive definite matrix $C\in\mathbb{R}^{n\times n}$, there exists an unique positive ...
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Stability of a sum of an unstable matrix and a stable matrix

I remember reading somewhere, probably in StackExchange, but now I couldn't find it, about the sum of two real matrices of dimension $n\times n$ $$ A + \theta B $$ where $A$ is non-Hurwitz, $B$ is ...
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Matrix exponential not going zero

Let $A \in \Bbb R ^{n \times p}$ be a real-valued matrix. What is the necessary and sufficient condition on $A$ such that the matrix exponential $$\exp( - tA^TA)$$ goes to zero as $t$ approaches ...
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How can I choose elements of a matrix to ensure it is a Hurwitz matrix?

I'm currently trying to implement an algorithm I found in a paper on occupancy detection. I have to "choose" values of $L_1$ and $L_2$ such that the matrix below would be a Hurwitz matrix. ...
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Check if block matrix is Hurwitz

How can I conclude from that fact, that $K_0$ and $K_1$ are positive definite diagonal matrices, that $$ A = \left( \begin{matrix} 0 & I \\ -K_0 & -K_1\end{matrix} \right) $$ is Hurwitz? Here, ...
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Find diagonal matrix $D$ such that $A D$ is Hurwitz

Let $A \in \mathbb{R}^{m \times m}$. Give necessary and/or sufficient conditions for the existence of a matrix $D \in \mathbb{R}^{m \times m}$ such that all eigenvalues of $AD$ have negative real part ...
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Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes the spectral radius of a matrix and the set of ...
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Is this a Hurwitz matrix?

I have the following block matrix $$J = \begin{bmatrix}A & B \\ C &D\end{bmatrix}$$ where $A, B, C, D$ are $2 \times 2$ matrices and 1.$A$ is Hurwitz and $a_{ii} < 0$ $B = \operatorname{...
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Randomly generate Hurwitz matrices?

Recall that a Hurwitz matrix is one whose eigenvalues lie in the left half plane; strictly Hurwitz such that they are in the strict left half plane. Is there way to randomly generate Hurwitz matrices?...
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Hurwitz matrix and positive definiteness

Hurwitz Matrices require that for a polynomial to have all negative real part roots then the determinant of the principle minors must be positive. I know that if we have a positive definite ...
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Condition for eigenvalues to have negative real parts (Hurwitz) for specific matrix structure

Let $$ A=\begin{bmatrix} P & \alpha x\\ -y^\top & 0\end{bmatrix}$$ where $P \in \mathbb{R}^{n \times n}$ is Hurwitz (the eigenvalues of $P$ have strictly negative real parts), $x, y \in \...
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Common quadratic Lyapunov function with all convex combinations Hurwitz

I have two $n\times n$ Hurwitz stable matrices, so $A_i\in H_n$ for $i=1,2$. I also know that every possible convex combination of these two matrices is Hurwitz, i.e., \begin{equation} C(A_1,A_2) = ...
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the largest real part of the eigenvalues of a matrix

I recently had an interesting observation but fail to come up with a rigorous explanation. What I observed is stated as follows. I am wondering if anyone is familiar with it. Let $A$ be a real ...
John Nan's user avatar
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Is a product of a Hurwitz matrix and a diagonal positive definite matrix always Hurwitz?

If I multiply a Hurwitz matrix (real part of eigenvalues are negative) with a diagonal positive definite matrix, does the product still remain as Hurwitz matrix?
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Convexity of the set of Hurwitz matrices

Determine whether the set of matrices $A\in \mathbb{R}^{n \times n}$ which are Hurwitz, denoted here by $\mathcal{H}$, (i.e., all their eigenvalues lies in the open half-plane $\mathbb{C}^-$) is ...
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Does subtracting a positive semi-definite diagonal matrix from a Hurwitz matrix keep it Hurwitz?

I am having a linear algebra problem here. I will be grateful if someone can help me. Let $A\in \mathbb{R}^{n\times n}$ be Hurwitz and diagonizable, and let $B$ be a diagonal matrix whose diagonal ...
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Is a Hurwitz matrix minus a positive-definite matrix still a Hurwitz matrix?

In a certain multi-agent systems, the $i$th system can be described by the form \begin{equation} \begin{array}{cc} \left\{ {\begin{array}{l} {{\dot x}_i}=A_i{x_i} +B_iu_i\\ y_i = C_ix_i ...
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Is this block matrix Hurwitz?

Let $A\in\mathbb{R}^{n\times n}$ be Hurwitz. Let $k_1,k_2>0$. Consider the matrix $$ M = \begin{bmatrix} 0 & I\\ k_1 A & k_2 A \end{bmatrix}, $$ where $0$ and $I$ denote respectively the ...
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Norm-bound for triangular Hurwitz matrix exponential

Let $(-A)$ be a real Hurwitz lower-triangular matrix (this implies that all the eigenvalues of $A$ are real and negative). Since $(-A)$ is Hurwitz, we know that there exist $\alpha,\lambda>0$ such ...
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Can the matrix product $PA$ be skew-symmetric with $P=P^T>0$ and $A$ Hurwitz?

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ is a real symmetric positive definite matrix. What will be ...
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Design feedback control law to make the whole matrix Hurwitz

Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant ...
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Is this matrix block matrix Hurwitz for some $c_1, c_2, c_3 > 0$?

Given the next well partitioned real squared matrix $$ M = \begin{bmatrix}A & \frac{c_3}{c_1}BC^T \\ c_3c_2C & E- c_3^2\frac{c_2}{c_1}CC^T\end{bmatrix}, $$ where $A$ is Hurwitz (all the ...
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If $A$ Hurwitz, $(A+A^*)$ is Hurwitz?

If I have $A$ Hurwitz matrix, is $(A+A^*)$, with $A^*$ the transpose of $A$, still Hurwitz? Any reference or proof? Because if $(A+A^*)$ is still Hurwitz I can say that it is even negative definite ...
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Checking if one "special" kind of block matrix is Hurwitz

I have the next block matrix $$ J = \begin{bmatrix}A & B \\ K &0\end{bmatrix} $$ all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...