Questions tagged [matrix-exponential]

"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."

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Multiplicative Identity for Exponential Function of Quadratic Matrix Form

We know that it is true for the exponential scaler manipulation $$e^{\frac{a + b}{c}} = e^{\frac{a}{c}}e^{\frac{b}{c}}.$$ However, suppose $\textbf{A} \in \mathbb{R}^r$, $\textbf{B} \in \mathbb{R}^r$ ...
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39 views

How to prove that $A^{p/q}=(A^{1/q})^p$?

defining matrix exponentiation for natural numbers by repeated multiplication and defining it for $\frac{1}{n}$ by: $A^{\frac{1}{n}}$ is the matrix s.t. $(A^{\frac{1}{n}})^n=A$. for a rational number $...
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54 views

matrix expression for $e^{iA}Be^{iA}$ in terms of anticommutators?

I'm familiar with the expression $$e^{-iA}Be^{iA} = \sum_{n=0}^{\infty} \frac{i^n}{n!}[..[B,A],\dots A]_{n \; \rm times}$$ for square matrices $A$ and $B$ and was wondering if equivalently $$e^{iA}Be^...
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Jacobian Matrix and Exponential Map?

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Can $f$ always be represented as $$ f(x)=\exp(F(x))x, $$ where $F:\mathbb{R}^n\to Mat_{n\times n}$? Intuition/Direction Somehow it seems ...
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Equation with matrix exponential integral

For my thesis I came across an equation that involves a matrix exponential within an integral, i.e. $ \int_0^{\Delta t} e^{-Ks}\Sigma \Sigma'e^{-(K)'s}ds = Q$ Where K is non-symmetric with real ...
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1answer
19 views

Semi-positivity of an Hermitian matrix

I´m triying to show that an $n\times n$ matrix $M$ such that $M_{ij}=e^{\gamma_{ij}}$ where $\gamma_{ji}=\gamma_{ij}^{\ast}$ are complex numbers is a positive semi-definite matrix. I made a proof of ...
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1answer
37 views

Find input for a control system such that system reaches a desired state $x_L$ at time $t=L$.

Given is the equation in state space of a control system $$\dot{x}(t)=Ax(t)+Bu(t)$$ $x(t)$ is the state vector of length $n$, $A$ is a square matrix of $n\times n$, and $B$ is a vector of length $n$. ...
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1answer
93 views

Is matrix exponential injective and surjective?

Exponential of real numbers exp: $\mathbb{R}\rightarrow\mathbb{R}$ is injective. Does the same hold for exponential of real matrices exp: $\mathbb{R}^{2\times2}\rightarrow\mathbb{R}^{2\times2}$? What ...
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On proving the surjectivity of the exponential map for compact Lie groups using properties of their Lie algebras [duplicate]

It is known that the exponential map is surjective for compact, connected Lie groups. This is generally proved through the introduction of some bi-invariant metric etc. etc. My question is related to ...
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3answers
55 views

Let $dX/dt=AX$. Find $e^{At}$ where $A=\left[\begin{smallmatrix} 1&1\\0&1\end{smallmatrix}\right]$

I have solved a couple of matrix exponential problems using $PDP^{-1}$ method. However, I am stuck on this problem as it is impossible to find the inverse of a singular matrix. The problem is :cLet $...
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1answer
25 views

Rewriting exponential function for a se(3) with application to computer perception algorithm

Problem $SE(3)$ is the group of isometries in $R^3$ and $se(3)$ its lie algebra. Let $M \in SE(3), \Delta m \in se(3), n \in se(3)$. Consider the function $f(\Delta m) = M \exp(n + \Delta m) $. Is ...
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21 views

Taking the “exponential of a distribution”

In Stroock's Probability Theory: An Analytical Perspective (great book, by the way), he says If $\nu$ is a probability measure on $\mathbb{R}^n$ and $\alpha \in [0, \infty)$, then the Poisson ...
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1answer
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Diagonalize an unknown matrix given only its eigenvectors and eigenvalues

The question is the following: The only way I can think of in doing this question would be for me to set variables for all values of A, and then using the given Eigenvectors and values to solve for ...
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Does the exponential map of a Lie Group satisfy a universal property?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $exp: \mathfrak{g} \to G$ be the exponential map. The map $exp$ gives a diffeomorphism of a neighborhood of the identity $0_{\mathfrak{g}...
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70 views

Proof of the derivative of a matrix exponential $\frac d {dt} e^{tA} = Ae^{tA}$

I have the proposition in my book that $$\frac d {dt} e^{tA} = Ae^{tA}$$ The proof provided is somewhat terse. I think I've proved it using games with indices, but the book's preferred proof uses ...
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21 views

Is the element of the exponential of a matrix equal to the exponential of the element?

I'm wondering if $(e^{A})_{ij} = e^{A_{ij}}$ I should prove that $(e^{A})_{ij} = \left(\sum\limits_{k = 0}^\infty \frac{A^k}{k!}\right)_{ij} = (\mathbb{I} + A + \frac{1}{2}A^2 + ...)_{ij} = \...
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The question about matrix exponential

Question: Explain why each of the matrices is NOT $e^{At}$ for any matrix A. $\begin{pmatrix} 0 & tan(t)\\ e^t & 1\end{pmatrix}$, $\begin{pmatrix} e^t & e^{2t}\\ e^{3t} & e^{4t}\end{...
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One-parameter subgroups of $GL_N(\mathbb{C})$ with measurable hypothesis is exponential map.

Let $T:(\mathbb{R},+)\to (GL_n(\mathbb{C}),\times)$ be a group homomorphism which is also a measurable (Lebesgue) function. The question asks to show that there exists a matrix $A$ such that $T(t)=\...
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Difficulty with a tricky matrix exponential step

I am having great difficulty checking a step involving operations with 2x2 matrix exponentials. The expression I would like to simplify is $$\lim _{x\to \infty} e^{-iHt-Vt}$$ where, for some $\...
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2answers
51 views

Find $A$ such that $e^A = B$ for a diagonal-matrix $B$

I'm trying to decide where're there exists a matrix $A \in \text{Mat}(2, \mathbb{R})$ such that $$e^A = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} =: B$$ In particular I don't want to use ...
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Find logarithm of matrix in diagonalform

I'm trying to decide whether there is a matrix $A \in \text{Mat}(2, \mathbb{R})$ with $$e^A = \begin{pmatrix}1 & -1 \\ -1 & 1\end{pmatrix}$$ I was already able to diagonalise the matrix into ...
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Matrix exponential from Lie adjoint but incremented factorial k

I have run into the following problem: $$ \sum_{k=0}^{\infty}\frac{1}{(k+1)!}(ad_{tA})^k(B) $$ Now if the factorial was $\frac{1}{k!}$, the result would be the conjugation of $B$ as $e^{tA}Be^{-tA}$, ...
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2answers
32 views

Getting the matrix exponential of a 2x2 with eigenvalues with real and imaginary parts

Struggling to simplify the matrix exponential of the following matrix: $$A = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}$$ It's trivial to observe the eigenvalues are $a±bi$, but the ...
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20 views

how to solve for 5 unknown exponents using logarithm

Hi all i am trying to determine the $\beta_1$ to $\beta_5$ for the following equation: $Y/B$ = $\beta_1[(H/B)^{\beta_2} * (\beta_1)^{\beta_3} * (\theta)^{\beta_4}]^{K^{\beta_5}}$ Where Y,H,B,$\...
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17 views

How to prove the infinity norm of soluton of the homogeneous heat equation decays with time?

For the heat equation $u_t = \mathcal{L} u, x\in [0, L]$ with $\mathcal{L} = \partial_{xx}$ being the Laplacian operator, if the initial solution is $u_0$, we can have the formal solution \begin{align}...
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24 views

A matrix exponential identity

In a physics textbook, the following identity was claimed to hold $$ e^{A+B} = e^{A} + \int_0^1 e^{\lambda(A+B)} B e^{(1-\lambda)A} d\lambda $$ with $A$ and $B$ both being complex matrices which do ...
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How to solve linear recurrence relation based on conditions?

Given $F(n)=a\cdot F(n-1)-b$ ,$n$ even $F(n)=a\cdot F(n-1)+b$ ,$n$ odd $F(0),a,b$ are constants. How to calculate $n$-th term in $log(n)$ time? I learnt matrix exponentiation technique. But ...
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20 views

Differentiability of matrix logarithm

Suppose $U\subset\mathbb{R}^n$ is open and $M:U\rightarrow \mathrm{GL}(m;\mathbb{C})$ is a differentiable matrix function. I would like to show the following statement: Let $x_0\in U$. Then there ...
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1answer
44 views

A matrix whose off-diagonal entries are $>0$ has its exponential with all positive entries

I am taking a course in differential equations and while I was doing some exercises I came across with the following statement: Let $A=(a_{ij})$ be a matrix such that $(a_{ij})>0$, $i\neq j$, then ...
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33 views

Relationship between use of logarithm map for lie groups as a coordinate chart and as a map from the lie group to the lie algebra

I am reading through a first course in differential geometry, and would like some clarification on something. I am aware of the exponential map mapping from the lie algebra to the lie group, close to ...
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52 views

Is the trace of the matrix exponential convex or non-convex?

I am trying to understand whether the following expression is a convex function: $$ f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\...
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1answer
66 views

Taylor-like bound for the matrix exponential tr(exp(X))

I am trying to prove the following Taylor-like bound for the the matrix function $F(X) := \operatorname{tr}(\exp(X))$, and symmetric matrices $X, V$ with $\lVert V \rVert \leq 1/2$: $$ F(X+V) \leq F(X)...
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Given the following matrix $M$, how can I find an equation for $M^n$?

Given the following matrix $M$, find an equation for $M^n$ $$M = \begin{pmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ 0 & 0& 0& 1 \\ 1& 0& 0& 0\end{...
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Proper- vs. pseudo-exponential maps to SE(3) and use cases for both

Background I am working on solving large-scale Bundle Adjustment in the context of robotics and Simultaneous Localization and Mapping (SLAM). This involves optimizing over camera poses (rotation and ...
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The exponential map on a Lie algebra which is a direct sum

I am asking the following question as I am trying to better understand Lie theory and the exponential map Assume $\mathfrak{g}$ is a (finite-dimensional) Lie algebra of the (matrix) Lie group $G\...
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System of exponential equations subject to differential inequality

Consider the following system of equations. $ \vec{y} = \vec{c}e^{\beta \vec{x}} $ where $\beta$ is a square matrix ($n x n$) such that $\beta_{ij} \lt 0 $ if $i=j$ and $\beta_{ij} \geq 0 $ if $i \...
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Why $[\mathbb R^d, \mathbb R^d] $ is the orthogonal group of $d$ dimension?

Please clarify the following question to from the book of P.Friz and N.Victoir intitled Multidimensionnal Stochastic Processes as a rough Paths Examples and Application on page 9: Example 5 Note ...
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1answer
136 views

Matrix Exponential of Sum of Matrices

I'm having trouble understanding why, in general, if $\mathbf A, \mathbf B$ are $n \times n$ matrices, it does not hold that $\exp(\mathbf A + \mathbf B) = \exp(\mathbf A)\exp(\mathbf B)$, but holds ...
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130 views

If $e^{A+B} = e^A e^B$, then $[A,B]=0$? [duplicate]

If two matrices $A$ and $B$ commute, then $e^{A+B} = e^A e^B$ by rearrangement of the $A$'s and $B$'s in the sum. But would the converse be true? So far I've tried to find a counterexample by ...
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64 views

Exponential of the differential operator $x \frac{d}{dx}$

I am studying conformal field theory, and I have run into an problem with calculations involving the quantum-mechanical translation and dilatation operators. It actually boils down to an issue about ...
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40 views

Computing matrix exponential

Prove that $$\exp\left(-\sum_{k=1}^{n-1}\frac{N^k}{k}\right)=I_n-N$$ for $\lambda \in \mathbb{C}$ and $$N=\left(\begin{array}{cccc}{0} & {\lambda} & {0} & {0} \\ {0} & {0} & {\...
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2answers
63 views

Linear ordinary differential equations and matrix exponentiation

I feel the solution to this problem is simple - but I am not entirely clear on what the question actually wants. Given the following: $$A = \left[ \begin{array}{ccc} -0.1005 & -0.266\\ -0....
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3answers
76 views

Logarithm of $I_n-N$ for $N$ nilpotent

Prove that $$L=-\sum_{k=1}^{n-1} \frac{N^{k}}{k}$$ is a logarithm of the matrix $I_n-N\!$, where $I_n$ denotes the identity, and $N$ is nilpotent such that $N^n=0$. More precisely, I want to show ...
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2answers
53 views

Are NumPy's choices for log of identity matrix and exponent of zero matrix correct?

Python's NumPy gives $$\log{\Big( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \Big)} = \Big( \begin{matrix} 0 & -\infty \\ -\infty & 0 \end{matrix} \Big)$$ and $$\exp{\Big( \begin{...
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1answer
31 views

Jacobian of rotation matrix w.r.t. SO(3) tangent vector

Trying to implement an Extended Information Filter and I have difficulty computing the Jabobians related to the below: Given a $3 \times 3$ rotation matrix $\mathbf{R}$ and the SO(3) logarithmic map ...
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0answers
20 views

Integrating $\mathrm{C}\cdot\exp(i\mathrm{A}x-\mathrm{B}x)\cdot\mathrm{1}$

The following signal is defined: $I(\omega) = real\{ \int_0^\infty \mathrm{W} \cdot \exp \{i(\mathrm{\Omega} -\omega \mathrm{E})t -\mathrm{K}t + \mathrm{R} t\} \cdot 1 dt\}$ Where: $ E = \begin{...
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1answer
57 views

Solve ODE system with generalized eigenvectors

So I’m trying to solve the system $x’=Ax$ with the initial conditon $x(0)=v_4$ with $A$ a 4x4 matrix with constant coeficients. And I am given the following properties; $$Av_1=2v_1\,;Av_2=-3v_2\,; ...
3
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1answer
109 views

Derivative of matrix exponential $\exp(A+xB)$ at $x=0$

Consider two (Hermitian) matrices $A$ and $B$. Is there a nice expression for the following? $$ \boxed{ \frac{\mathrm d}{\mathrm d x} \exp\left( A + x B \right)\big|_{x=0} = \; ? }$$ Of course, if ...
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1answer
88 views

Prove $2 \times 2$ real matrix is not the exponential of some other real matrix

How do I prove that the real matrix $A$, where $$ A=\begin{pmatrix} -3 & 0\\ 0&-5 \end{pmatrix}$$ can not be written es the exponential of another real matrix?
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19 views

Conditions for $e^X$ to be an orthogonal matrix [duplicate]

If $X$ is a skew-symmetric matrix, its exponential $e^X$ is orthogonal. Does the converse implication hold?

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