Questions tagged [matrix-exponential]
"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."
711
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Differentiating $\exp^{DA}$ w.r.t. the diagonal matrix
Let $D$ be a diagonal matrix
$$
D = \begin{bmatrix}
d_1 & 0 & \dots & 0 \\
0 & d_2 & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & ...
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2
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Derivative of a matrix times a vector within an exponential function
Let us consider $\beta:=(\beta_1,...,\beta_p)^T$ and $X$ a matrix of dimension ($n\times p$). I would like to calculate the following derivative $$\frac{\partial}{\partial \beta}\exp(X\beta) $$
Is it ...
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Differentiate exponential of a matrix times a vector
I have:
$\boldsymbol{r}$, a $(1 \times N)$ vector of times
$\gamma$, a $(p \times 1)$ vector of regression parameters
$V$, a $(N \times p)$ matrix of covariates
I want to get the expression to ...
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27
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Discrepancy in Calculating Stationary Distribution of a Markov Chain
I'm studying a Markov chain problem question 15 where a rat runs through a maze with the following transition matrix P:
...
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Exponentiate rate matrix for continuous time Markov chain describing elementary transitions between subsets of {1, ..., n}.
My continuous time Markov chain describes transitions between subsets of ${\{1,\dots, n\}}$. Only elementary transitions are considered (+1 element, -1 element), and the empty subset is absorbing. The ...
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1
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Solve matrix differential equation around known solution
Define a matrix differential equation $\dot{X}=A(t)X(t)$ with initial condition $X(0)$, where $X=[x_1,x_2,\ldots]^T$ is a 1D vector and $A(t)$ is a complex-valued time-dependent matrix.
This system ...
- 99
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When $\text{exp}(A) = B$ has a solution
Let $e^A$ denote the matrix exponential, for $n \times n$ matrices over $\mathbb{C}$. I am trying to find for which $B$ there exists a solution $A$ to the equation $e^A = B$.
Clearly a necessary ...
- 384
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exponential of matrix product
Is there any way to reformulate exponentials of matrix products ($e^{XY}$), where $X,Y\in \mathbb R^{n\times n}$? I am interested in how $e^X, e^Y$ relate to $e^{XY}$.
Of particular interest to my ...
- 135
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answers
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$n$-th derivative of $e^{Y(\delta,t)}$ at at $\delta=0$ for a special $Y(\delta,t)$
Posting again to get better traction.
I am looking for a representation of the expression:
$$
\frac{\partial^n} {\partial\delta^n} e^{Y(\delta,t)},
$$
where $Y(\delta,t) : \mathbb R^2_+ \rightarrow \...
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Equation with Matrix exponential
i am working on a paper regarding Mortality Modeling. I have the following equation and I can't figure out why the following equation holds. I think the background knowledge is not important, it's ...
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Constructing diagonalizable matricies such that matrix exponential of their product is easy to compute.
Question Consider a real skew-symmetric matrix $J$ and a real symmetric positive semi-definite matrix $A$. We want to parameterize a construction of these matrices so that it is easy to compute $e^{JA}...
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1
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simplifying trace of product of two unitary operations
For Hermitian matrices $A$ and $B$, diagonalizing each unitary matrix $e^{iA}$ and $e^{iB}$ gives
\begin{equation}
\text{Tr}(e^{iA}e^{iB}) = \text{Tr}(U^* e^{iD_A}UV^*e^{iD_B}V),
\end{equation} where $...
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Exponential map of flow, does it locally look like a power series/matrix exponential?
Let $g_t : M \rightarrow M$ be the flow of a vector field, that is a family of diffeomorphisms such that $g_t \circ g_s = g_{t+s}$ to $t,s > 0$ and $g_0 = \operatorname{id}$.
One can interpret the ...
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Express $\exp(A+B(t))$ as $\exp(A) + \cdots$
Consider two $n\times n$ matrices, $A$ and $B(t)$. Here $A$ is time independent, and $B(t)$ is time dependent such that $A$ and $B(t)$ don't necessarily commute.
I want to analyze $\exp(A+B(t))$. ...
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Help me understand this problem please
As you can see in equation 15.19, the authors of the book replaced $ (-a R_t +1) $ with the $ \exp(h)$ with $h =2.5$.
Can you help me understand how the authors got this value? I searched the whole ...
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answer
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Differential of a matrix exponential
I need your help with the differential of a matrix exponential.
A function $f$ is a mapping from a matrix to a matrix, that is, $f: \mathbb{R}^{n \times n}\rightarrow \mathbb{R}^{n \times n}$.
Here $\...
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Linear second order homogeneous matrix ODEs with constant coefficients: Solution strategies?
$\newcommand{\bm}[1]{\boldsymbol{#1}}$
$\newcommand{\img}{\operatorname{img}}$
The Scalar Setting
When looking for a solution $u:\mathbb{R}\to\mathbb{R}$ of the following linear homogeneous second ...
- 2,027
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Upper bounding $\|\sum_{k = 0}^{N} \frac{(At)^k}{k!}\|$
I'm trying to figure out the tightest upper bound for
$$\left \|\sum_{k = 0}^{N} \frac{(At)^k}{k!} \right\|,$$
where $A$ is a matrix and $\|A\|t > 1$. I know that $e^{\|A\|t}$ is a known upper ...
- 43
2
votes
1
answer
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Evaluation of exponential matrix integral [closed]
Let $A$ and $B$ be symmetric $n\times n$ matrices, and let $D$ be a diagonal $n\times n$ matrix. Assume that all matrices are invertible. In my particular case of interest $A=B$ but this may be ...
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ODE with tridiagonal anti-Hermitian matrix and initial condition in one entry
I am given an anti-Hermitian and tridiagonal matrix $\bf M$ and I need to solve for a vector $\bf v$ such that
$$ \left[ \exp( t {\bf M} ) \, {\bf v} \right]_{1} = {\bf 0} $$
or at least $\approx {\bf ...
- 33
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Space-variant diffusion with infinite speeds: eigendecomposition and matrix exponential
The heat diffusion equation on some domain $\Omega$ with Neumann boundary conditions on $\partial\Omega$ and normal $n$ is given as:
\begin{alignat}{3}
\partial_t u(t,x) &= \Delta u(t,x), &\...
- 2,027
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1
answer
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views
Finding the matrices $J$ (Jordan normal form) and $T$ so that $J=T^{-1}AT$
To solve an automation engineering exercise, I need to find the Jordan normal form $J$ and a matrix $T$ so that $J=T^{-1}AT$, where $A$ is the initial matrix (given by the exercise). For example:
$$A =...
1
vote
1
answer
83
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On the weighted adjacency matrix of a directed acyclic graph (DAG)
A matrix $W \in \mathbb{R}^{d \times d}$ is the weighted adjacency matrix of a directed acyclic graph (DAG) if and only if
$$ h(W) = \operatorname{tr} \left( \exp(W \circ W ) \right) - d = 0 $$
where ...
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Positivity of semigroup $(P_t)_{t\geq0}$ implies its contractivity, $\| P_tf\| \leq \|f\|$
Let $\Omega$ be a Polish space, denote $E = (C_b(\Omega),\|\cdot\|)$ be the Banach space of continuous and bounded functions with the usual supremum norm. A family $(P_t)_{t\geq 0} $ of linear ...
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Re-arranging products of exponentials for a Lie algebra
Suppose I have $n$ operators $X_1,\ldots,X_n$, which are closed under commutators:
$$[X_i,X_j]=c_1X_1+\cdots+c_nX_n,$$
where $c_j$ are constants (which will depend on $i$ and $j$).
Now consider a ...
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votes
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How can one obtain the resulting 3D rotation matrix by "integrating" over some time-dependent angular velocity vector?
This question comes from my own thinking, so I want to check with others to make sure I'm not led astray.
Let $R(t): [0, \infty)\rightarrow\textrm{SO}(3)$ be a differentiable path starting at $R(0) = ...
- 2,427
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0
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27
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Exponential of an integrated matrix
I am starting to read about Floquet theory and its basic setup
$\dot{x} = A(t) x$
where $A(t)$ is a $N$-dimensional $T$-periodic bounded square matrix for all $t$ and $x \in \mathbb{R}^N$.
I was ...
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votes
1
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Matrix of a given Operator and a little identity about powers of a certain matrix type
I have asked a long time ago a question about matrix exponential shifting. I considered the following two operators $D$ and $S$ in the vector space $\mathcal{P}_{n}$ of polynomials of degree $\leq n$ :...
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views
Are the exponential and matrix exponential valid over an arbitrary field $\mathbb{F}$?
I understand that if $[A] \in M_{n\times n}(\mathbb{C})$ then the matrix exponential of $[A]$ is denoted $e^{[A]} \in M_{n\times n}(\mathbb{C})$, and is defined by
$$e^{[A]} := \sum_{k=0}^\infty \frac{...
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$3\times 3$ Matrix exponent [duplicate]
I have a problem where I’m given the matrix
$$
B =
\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 1\\
0 & 0 & 1
\end{pmatrix}.
$$
I’m tasked with computing $e^B$.
Now the point where I’m ...
0
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5
answers
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views
Show that $\exp \begin{pmatrix} x & -y\\ y & x\end{pmatrix}= \exp(x) \begin{pmatrix} \cos y& -\sin y\\ \sin y& \cos y\end{pmatrix}$
Show that $$\exp \begin{pmatrix}
x & -y\\
y & x\end{pmatrix}= \exp(x) \begin{pmatrix}
\cos y& -\sin y\\
\sin y& \cos y\end{pmatrix}$$
for all $ x,y \in \mathbb{R} $.
My thought ...
- 1,313
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votes
2
answers
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Bounding error of approximating $(I-A)^t$ with $\exp(-At)$ for large $t$
Suppose $A$ is a diagonal matrix with diagonal entries $\in (0,1]$
I'm interested in bounding the following quantity in terms of $t$:
$$f_A(t)=\operatorname{Tr}\exp(-At)-\operatorname{Tr}(I-A)^t$$
...
- 4,873
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votes
1
answer
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views
Using $\exp(-At)$ to approximate $(I-A)^t$
Suppose $A$ is a $d\times d$ positive definite convergent matrix and $t>1$. How do I upper bound the following in terms of simple statistics of $A$?
$$f_A(t)=\operatorname{Tr}\exp(-At)-\...
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0
answers
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views
How to find the Fréchet derivative of a matrix exponential?
Let $f : \Bbb R^{n\times n} \to \Bbb R^{n \times n}$ be the matrix exponential $$ f(A) = \sum_{k = 0}^\infty \frac{A^k}{k!} $$ Let $B$ be a matrix that commutes with $A$. Show that the value of ...
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Application of a basis of the Lie algebra on the Lie exponential
Let $G$ a compact connected (matrix) Lie group and $\mathfrak{g}$ its Lie algebra. Let $\{E_i\}$ be a basis of the Lie algebra thought as a vector space. Any element of the Lie group $G$ can be ...
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Are there finite dimensional matrices for which $e^{A+B}=e^Ae^B$ when $A$ and $B$ do not commute?
If we have square matrices $A$ and $B$ that commute (i.e. $AB=BA$), then we have $e^{A+B} = e^Ae^B$. In general this isn't true without the condition that $A$ and $B$ commute. I would like to know if ...
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votes
2
answers
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views
Convexity of the norm of a matrix exponential
I would like to know the convexity of the function below.
Denote the space of $ n \times n $ real symmetric matrices as $\mathcal{S}^n$, then define $f:\mathcal{S}^n \rightarrow \mathbb{R}$ as
$$
f(X) ...
- 165
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votes
1
answer
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views
Computing the vector field, matrix exponential, normal forms (Rossmann)
In the book Lie Groups by the author Rossmann, if $X=\begin{bmatrix} 0 & -1\\ 1& 0 \end{bmatrix}$ , its exponential matrix is given by $\exp(t X)=\begin{bmatrix}\cos(t) & -\sin(t)\\ \sin(t)...
- 3,336
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Numerically stable and efficient way to compute matrix $Z$ such that $e^Z=e^Xe^Y$
Is there a known way to calculate matrix $Z$ such that $e^Z=e^Xe^Y$ in an efficient and numerically stable way? $X$ and $Y$ are square matrices and can be complex.
Naively computing it is numerically ...
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0
answers
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views
Under what conditions is the matrix exponential $ e^A $ symplectic?
Let $ A \in \Bbb R^{2n \times 2n}$ and let the matrix exponential $ e^A $ be defined by
$$ e^A := \sum_{k=0}^{\infty}\frac{A^k}{k!}.$$
I want to ask under what conditions the matrix $ e^A $ is ...
2
votes
0
answers
158
views
Logarithm map for Lie groups other than Matrix Lie groups
Hello Lie group experts!
My question is simple: Is there a construct for logarithm map of non-Matrix Lie groups? Besides; I believe it exists, because $exp(t.X) \in G \neq GL(n, F)$ being ...
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views
Solution of matrix exponential integral $\int_0^{2 \pi} d t \exp(A \cos(t) + B \sin(t))$ with non-zero commutator
What is the integral of the matrix exponential
$$\int_0^{2 \pi} dt \exp(A \cos(t) + B \sin(t))$$
with matrix $A$ and $B$.
The commutator $$\left[A,B\right]\neq0$$ is non-vanishing.
I approached the ...
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votes
1
answer
73
views
How to derive $Exp(\phi) R = R Exp(R^T \phi)$ for SO(3)
I am reading https://limhyungtae.github.io/2022-04-01-IMU-Preintegration-(Easy)-4.-Derivation-of-Preintegrated-IMU-Measurements/
And trying to work my way through it, I already asked some questions ...
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56
views
Can generating functions be used to solve evolution matrix differential equations and recurrence relations of matrices?
Generating functions seem to be a powerful tool in discrete mathematics for solving differential equations and recurrence relations. I've been trying to figure out if these methods can be expanded to ...
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votes
1
answer
57
views
closed solution for non linear system
For linear systems of the form $Ax = \dot x$, one can find a close solution of the form:
$
\dot x(t) = e^{At} x(0)
$
Is there a way to show that a nonlinear solution of the form, $A(x) x = \dot x$ has ...
- 59
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votes
1
answer
40
views
Largest Singular Value of the Exponential of a Normal Matrix
I am trying to work out a relation between the largest singular value $\sigma_1(Q)$ of a normal matrix $Q$, and the largest singular value of $e^{Q}$. In an ideal world for the wider context of my ...
- 11
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vote
0
answers
96
views
Calculation of the $3\times 3$ exponent matrix via Cayley-Hamilton theorem.
I have a random $3\times 3$ matrix $A$.
How can I calculate $e^A$ by $E$ (the identity matrix), $A$ and $A^2$, using the Cayley-Hamilton theorem?
I need a general expression that includes only the ...
- 11
3
votes
2
answers
240
views
Generalisation of Baker–Campbell–Hausdorff (Matrix exponential simplification)
Let there be some matrix exponential function for matrices of the form $$M=\sum_{i=1}^p c_i A_i$$ where $M,A_i\in\mathbb{R}^{n\times n}$ are given, real square matrices and $c_i \in \mathbb{R}$. In ...
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Can we use the singular value decomposition to compute the matrix exponential for a non-diagonalisable matrix?
For a diagonalisable matrix $ \bf{A} $ with eigendecomposition $ \bf{A} = \bf{U} \bf{\Lambda} \bf{U}^{-1} $, we know that $ \exp(\bf{A}) = \bf{U} (\exp \bf{\Lambda}) \bf{U}^{-1} $, where $ \exp \bf{\...
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How to differentiate a vector related to a matrix w.r.t. a vector by using Frobenius product notation
Here is the equation I want to solve:
$$\frac{d\vec{A}}{d\vec{C}}=\frac{d\mathbf{M}(\vec{C})\vec{B}}{d\vec{C}}$$
where
$\vec{A} = \mathbf{M}(\vec{C})\vec{B}$
$\vec{B}$ is a constant vector
$\vec{C}$ ...
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