A square matrix $A$ is a Hurwitz matrix if all eigenvalues of $A$ have strictly negative real parts.
Questions tagged [hurwitz-matrices]
38 questions
9
votes
1 answer
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 Hurwitz stable matrices is
\begin{align*}
\mathcal H =…
user1101010
- 3,638
- 1
- 17
- 40
8
votes
1 answer
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 part being $A = - (B + B^T)$, and $K$ is a diagonal…
user51196
6
votes
1 answer
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 definite matrix $B\in\mathbb{R}^{n\times n}$…
graham
- 336
5
votes
1 answer
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 being symmetric and with real eigenvalues negative.
Michele Tuloski Furci
- 179
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- 6
4
votes
1 answer
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), $x, y \in \mathbb{R}^{n}$, and $\alpha$ is a real positive scalar. Find…
neelarnab
- 189
3
votes
1 answer
Find all stable dynamical systems in 2 variables
- To set jargon: The set of all $2\times2$ real matrices is $\mathbb{R}^4$. The 4 matrix elements fully parametrise this set: if we see this set as a manifold, then the 4 parameters are coordinates (we can always change coordinates and find another…
Quillo
- 2,260
3
votes
3 answers
Is the linear interpolation of two stable matrices always stable?
I have a question regarding the stability of linear systems. Let's assume we have two stable linear systems represented by matrices $A_1$ and $A_2$, where both matrices have eigenvalues with strictly negative real parts (i.e., both systems are…
홍정국
- 49
3
votes
1 answer
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, $I$ is the identity matrix and the dimension of all…
Steradiant
- 203
3
votes
1 answer
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? I came up with two methods,
Randomly sample…
ITA
- 1,893
3
votes
0 answers
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 matrix then all the real parts of the eigenvalues are…
Matthew
- 1,384
3
votes
1 answer
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) = cA_1+(1-c)A_2 \in H_n \hspace{0.5cm} \forall…
user480735
- 41
3
votes
2 answers
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 elements are non-negative. Is $A-B$ still…
MLearner
- 33
2
votes
1 answer
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 the condition(s) (if at all) for the product…
neelarnab
- 189
2
votes
1 answer
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 eigenvalues have negative real part), $CC^T$ positive definite,…
user51196
2
votes
0 answers
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 determine whether all real parts of eigenvalues…
Kyle
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