# Tagged Questions

**4**

votes

**1**answer

41 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

**2**

votes

**0**answers

27 views

### When is the solution to a n initial value problem matrix differential equation invertible? [migrated]

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$:
$$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$
$$ ...

**7**

votes

**3**answers

360 views

### Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**4**

votes

**1**answer

98 views

### For a linear dynamic system, what can we learn from its singluar value and rank?

Given a linear system $\frac{dx}{dt}=Mx$, what's the relationship between the dynamic's property and the singular value decomposition/rank of $M$ ?

**3**

votes

**1**answer

192 views

### ODE in symmetric definite positive matrices

It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt ...

**4**

votes

**4**answers

976 views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**5**

votes

**1**answer

212 views

### Modeling free Lie algebras with matrix algebras

I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that ...

**5**

votes

**1**answer

563 views

### Decide how many non-negative solutions a set of multivariate quadratic equations have

Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?
Some explanations:
All the coefficients are real numbers.
The number ...

**1**

vote

**2**answers

274 views

### Using Wavelet Transforms to Approximate Matrices

It's a long time since I worked on this kind of problem, so please bear with me.
I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...

**2**

votes

**3**answers

267 views

### In an n-dimensional linear 2nd-order ODE, why is the transpose-inverse to a system of solutions also a solution?

I'm at a sticky spot in my research. Namely, I have a particular fact, and it ought to have a short proof, but the only way I know how to show it is long and drawn out, and I don't like it and worry ...