# 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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### Routh Array: Special Case

I've seen all texts say that "If only first element in any row of routh's array is zero, you replace it with a small positive number, Ephsilon. Now, if after solving the array there is no sign change ...
30 views

### Norm of Hurwitz-perturbed matrix

Recently I noticed a very unusual statement in the paper Konda,Tsitsiklis Convergence Rate of Linear Two-Time-Scale Stochastic Approximation 2004. It is in Lemma A.1 and stated as follows. Let B be ...
27 views

### Minimum and maximum rate of convergence of the dynamics involving a Hurwitz matrix!!

Let $R$ be a Hurwitz matrix, then there exists a unique positive-definite solution $P$ to the following equation \begin{equation} R^{T}P+PR=-I,\tag{1} \end{equation} where $I$ is the identity matrix ...
118 views

### 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?...
245 views

### 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 $P \in \mathbb{R}^{n \times n},\quad x,y \in \mathbb{R}^{n}, \quad \alpha$ is a real positive scalar, and eigenvalues ...
376 views

### 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 ...
305 views

### 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?
371 views

### 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 ...
158 views

### 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 ...
### 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 ...
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 ...