All Questions
Tagged with matrix-exponential linear-algebra
13 questions
6
votes
1
answer
239
views
Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 361
1
vote
2
answers
152
views
Property for bounding matrix exponential
Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
- 21
3
votes
1
answer
455
views
Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE
$$f(t)=\langle\exp(-...
- 2,442
4
votes
0
answers
163
views
Matrix logarithm of unitary factor from polar decomposition of product of positive definite matrices
This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post ...
- 41
1
vote
1
answer
136
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
8
votes
1
answer
2k
views
Matrix elements of exponential of tridiagonal matrices
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...
- 371
4
votes
0
answers
139
views
Is there a nice way to express a matrix exponential when rows are proportionally scaled?
Assume I am given an $n \times n$ matrix $A$ with real or complex coefficients. Its matrix exponential is denoted by $\exp(A)$ and is calculated as usual. Assume further that I want to rescale the ...
- 749
3
votes
1
answer
124
views
Higher order Lyapunov equation
Let $A$ be a (finite) Hurwitz matrix.
In this related question of mine, (see also https://en.wikipedia.org/wiki/Lyapunov_equation) it is shown that
$$
\int_0^\infty \sum_{j,k} (e^{At})_{ij} Q_{jk} (...
- 562
3
votes
2
answers
1k
views
Properties of matrix exponential without using Jordan normal forms
There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:
All eigenvalues of $A$ have negative ...
- 791
5
votes
3
answers
830
views
Integral of the entrywise square of the exponential of a matrix
Note: I posted my question on math.stackexchange but got no answer. That is why I am asking it here.
Let $A$ be a $n\times n$ square matrix such that the real part of all eigenvalues are negative. ...
- 562
12
votes
2
answers
9k
views
What is the time complexity of the matrix exponential?
While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...
- 241
9
votes
3
answers
4k
views
Fast Upper Triangular Matrix Exponentiation
Let $Q_n$ be a $n\times n$ matrix with $Q_n=\begin{pmatrix} -\lambda_1-\mu_1 & \lambda_1 & 0 & \cdots\\ 0 & -\lambda_2-\mu_2 & \lambda_2 & \cdots\\ \vdots & \vdots & \...
- 4,952
63
votes
7
answers
9k
views
How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
- 845