Questions tagged [matrix-exponential]

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

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Why does the matrix exponential $e^A$ always exist?

Why does $e^A$ always exist for any given $n \times n$ matrix $A$? I can't find anything discussing this question, which is quite suprising, since it is such a general question.
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Higher order derivative of exponential map

The derivative of the exponential map is given by (wiki): $$ \frac{d}{dt} e^{X(t)} = e^{X(t)} \frac{1 - e^{-ad_{X(t)}}}{ad_{X(t)}} \frac{d}{dt}X(t) $$ Is there a reasonable formula for higher order ...
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Lie bracket for $GL_n\mathbb{R}$ from the composition of two flows of left invariant fields

I'd like to understand the following passage from Arvanitoyeorgos' "An introduction to Lie groups and the geometry of homogeneous spaces", where the author explains why for any $A,B \in M_n\...
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Does $e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n$ hold for matrices?

Let $X$ be a $d \times d$ real matrix, $d>1$. Is it true that $$ e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n\,\,\,? $$ Edit: It seems that this question is a duplicate. To make it ...
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Matrix exponential of infinite antisymmetric matrix with entries only next to its diagonal

What is the exponential $\exp (t A)$ of the operator $A$ whose components are given by $A_{nm} = \delta_{nm-1} \sqrt{n+1} - \delta_{nm+1}\sqrt{n}$ where the $n,m \in \mathbb{N}_0$. If we just consider ...
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Converting recursive equation into matrices by using matrix exponentiation

This is an example of converting fibonacci function into matrices called matrix exponentiation method. Fibonacci sequence defines $$ f(1)=1 $$ $$ f(2)=1 $$ $$ f(x) = f(x-1) + f(x-2) $$ This recursive ...
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Proof the limit of matrices means about $e^H$ (where matrix $H$ is self-adjoint)

This problem is from Chap.6 of Introduction to Matrix Analysis and Applications of Petz. Prove for self-adjoint matrices $H$, $K$ that $$ \lim _{r \rightarrow 0} \left(e^{r H} \#_{\alpha}e^{r K}\...
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Why is the domain of the exponential function the Lie algebra and not the Lie group?

The exponential function as I know it is defined as: $$\exp:\mathfrak{g}\to G$$ and it gives each element $X$ the value of $\exp_X(1)$ where $\exp_X$ is the unique $\mathbb{R}\to G$ homomorphism that ...
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Why matrix exponential in two different methods not matching?

Consider the following matrix: $$A=\left[\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right]$$ We need to find $e^{At}$. Method $1.$ Th eigen values of $A$ are $3,-1$. I have Diagonalized the ...
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One-Parameter Subgroups in Hall B. (Theorem 2.14)

Theorem 2.14 (One-Parameter Subgroups) If $A(\cdot)$ is a one-parameter subgroup of $\text{GL}(n;\mathbb{C})$, there exists a unique $n\times n$ complex matrix $X$ such that $A(t)=\mathrm{e}^{tX}$ In ...
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Averaged matrix exponential inverse

Let $H$ be a given symmetric matrix, i.e., $H\in\mathbb{R}^{d\times d}_{\rm sym}$, $d\geq1$. The operator $T$ is given by $$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
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Using Jordan Form to understand $\dot{x} = Ax$

I know the flow for the differential equation $\dot{x} = Ax$, where $A$ has no time dependency, is given by $\phi(x,t) = e^{At}$. Since we want explicit representations we use Jordan Form and ...
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Calculating a matrix-exponential [duplicate]

Let A be the following matrix. $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ \end{pmatrix} $$ I have to calculate $e^A$. My idea was to diagonalize A because then $e^A = Pe^DP^-1$ if $A = PDP^-1$. But A ...
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Is the exponential of a linear operator T still a linear operator?

Consider a linear operator $T: \mathrm R^n \rightarrow \mathrm R^n $. Given: $$ e^T = \sum_{k=0}^\infty \frac{T^k}{k!} $$ Is $e^T$ still a linear operator? If it is, why? One should note that: Given ...
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Can systems of linear ODE's be solved when their matrix is not diagonalizable?

I'm working through my ODE homework right now and I've run into a repeated issue of ODE systems not being diagonalizable. I am not aware of any other methods to solve systems and my lecture notes do ...
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Zassenhaus formula for arbitrary (faithful) representation of SU(2)

In Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure a BCH formula is given which works for an arbitrary faithful representation of SU(2). Now, is there an equivalent ...
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Is $e^{itA}$ an open map?

Let $A$ be a diagonal matrix with real entries. I want to find out whether the exponential map $t \mapsto e^{itA}$ is open, where $t \in \mathbb{R}$. My observation: the map is injective since fixing $...
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What is the correct Taylor expansion of exponentiated operators?

I am trying to derive the following result from this paper (Equation S3):$$e^{-ik_z\hat{z}}\hat{\rho}e^{+ik_z\hat{z}} - \hat{\rho} \approx k_z^2[\hat{z},[\hat{z}, \hat{\rho}]]$$with operators $\rho, z$...
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Dealing with exponential operators

I have some troubles with the following equations. First, we define a function $g\left(\hat{\vec{x}}\right) = \exp[-(\vec{x}-\hat{\vec{x}})^2]$. We want to expand $g$ around $\hat{\vec{x}}\approx0$. ...
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Prove that the exponential of matrix is well defined

We just saw today the concept of the exponential of a matrix. Let $A\in \mathbb R^{n\times n}$ a matrix. How can I prove that $$e^A:=\sum_{k=0}^\infty \frac{A^k}{k!}$$ is well defined ? What I've done ...
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Find $\exp\bigg[t\left ( \begin{matrix} -1& 1 \\ 0 & -1 \\ \end{matrix} \right )\bigg]$ by definition

Find $\exp\bigg[t\left ( \begin{matrix} -1& 1 \\ 0 & -1 \\ \end{matrix} \right )\bigg]$ by definition Denote $A=\left ( \begin{matrix} -1& 1 \\ 0 & -1 \\ \end{...
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Are all the entries of the matirx exponential function non-negative? How to compute the infinity-norm of a matrix function?

The exponential functions are essential ingredients of Exponential Time Difference (ETD) integrators. Let us introduce following functions \begin{equation*} \varphi_k(z) = \int_0^1 \mathrm{e}^{(1 - s) ...
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Exponential matrix converge using bounded matrix

Given that entries of matrix $A^{n}$ is bounded by $r^{n-1}M^{n}$, how to use this property to prove the exponential matrix $e^{A}$ converges? It's clear that $\lVert A^{n}\rVert \le r^{n-1}M^{n}$ so ...
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Matrix exponential with derivative matrix (exponential shift)

For an operator $A: V \rightarrow V$ in the finite-dimensional vector space $V$, we define the exponential function $\exp (A)$ as the following operator in $V$. $$ \exp (A)=\sum_{k=0}^{\infty} \frac{A^...
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differential equation using matrix exponential not consistent solution

For the differential equation (physical friction) $\ddot x=-a\cdot \dot x$ The solution can be easily found using exponential ansatz and is $x(t)=c_1+c_2 \exp(-a\cdot t)$ Or expressing this using ...
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Complex matrices with real exponential

The set of all complex numbers $z$ such that $e^z$ is real, is exactly the set $\mathbb{R}+i\pi \mathbb{Z}$. My question is : can we have a similar result for matrices ? More precisely : is there a ...
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convergence to matrix exponential

I know that $$\left[1+\frac1n x +O\left(\frac1{n^2}\right)\right]^{tn}\to e^{tx}$$ as $n\to\infty$ for any real numbers $x,t$. And we get other variations that as additional perturbations to the ...
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Finding an operator C that satisfies AB=CA

Let $D=\frac{d}{dx}$ , $A=\sum_{i=-n}^{i=n} a_i(x)D^{i}$ and $B=b(x)D$, where $a_n(x)$ and $b(x)$ are sufficiently smooth functions and $n$ is an arbitrary positive integer. $A$ may not be invertible. ...
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Calculate ${e^{At}}$ of $A = \left( {\begin{array}{*{20}{c}} i&j&k\\ i&j&k\\ i&j&k \end{array}} \right)$ knowing that $i+j+k=0 $ [closed]

How to calculate ${e^{At}}$ for a matrix $A = \left( {\begin{array}{*{20}{c}} i&j&k\\ i&j&k\\ i&j&k \end{array}} \right)$ knowing that $i+j+k=0$ answer: if you calculate $A^2$ ...
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when does ${e^{\left( {A + B} \right)t}} = {e^{At}}{e^{Bt}}$? [duplicate]

is it sufficient that $AB=BA$ to conclude that ${e^{\left( {A + B} \right)t}} = {e^{At}}{e^{Bt}}$ ?
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2 answers
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Compute ${e^{At}}$ for the matrix $A=\left[ {\begin{array}{*{20}{c}} B&I\\ 0&B \end{array}} \right]$

What is the easiest approach for computing ${e^{At}}$ for the following block matrix? $$A = \left[ {\begin{array}{*{20}{c}} B&I\\ 0&B \end{array}} \right]$$ my attempt: I was trying to write ...
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2 answers
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Find $e^{xA}$ by definition

$A =\left [ \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{matrix} \right ] $ I have to find ...
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Matrix exponential $\exp(tJ)$: Is this a mistake in the solution to this exercise?

I just solved another problem but my result is different from the solution and I suspect this could be a mistake in the solution. Can anyone confirm that this is a mistake in the solution? The matrix $...
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Relationship between Fourier transform and matrix exponential

Let's assume we have (in vector form) $$\dot x(t) = A x(t)+L w(t)$$ The solution to this general linear time-invariant non-homogeneous ODE, with null boundary values, can be written as $$x(t) = \int^...
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One-parameter subgroups of $GL_n(\mathbb{C})$ passing through a given real matrix

Let $\gamma$ be a matrix in $GL_n(\mathbb{R})$. Then $\gamma$ lies on a 1-parameter subgroup of $GL_n(\mathbb{C})$ even though it may not lie on a 1-parameter subgroup of $GL_n(\mathbb{R})$ (for ...
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For what $X \in \mathfrak{su}(N)$ do we have $e^{2\pi X}=1$?

The Lie algebra $\mathfrak{su}(N)$ consists of skew-Hermitian $N\times N$ matrices. It has an $\mathbb R$-basis $\{X_{ij}, Y_{ij}\}_{1\leq i<j\leq N} \cup\{Z_i\}_{1\leq i \leq N-1}$, where $X_{ij}$ ...
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$A=\begin{pmatrix}5 & 2 \\ 4 & 7\end{pmatrix}$ ,find $e^{xA}$

$A=\begin{pmatrix}5 & 2 \\ 4 & 7\end{pmatrix}$ I have to find $e^{xA}$ by definition , $A^0=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ $A^1=\begin{pmatrix}5 & 2 \\ 4 & 7\end{...
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1 vote
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Calculating an element $b_{ij}$ of matrix exponential $(b_{ij})=B=\exp(A)$

Is there some easy way to calculate an element $b_{ij}$ for fixed $i$ and $j$ of the matrix exponential $(b_{ij})=B=\exp(A)$? ($A$ is symmetric real matrix) I know that I could diagonalize $A$ and ...
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2 votes
1 answer
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Matrix exponential via Cayley-Hamilton

Problem For any $t\in\mathbb{R}$ compute $\exp(A_\omega t)$, where \begin{equation*}A_\omega\triangleq\left[\begin{array}{c|c} 0_2 & I_2 \\ \hline 0_2 & \Omega \end{array}\right]\end{equation*}...
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1 vote
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Solving a parameterized matrix equation with specific structure

Consider the equation: $\mathbf{y} = \mathbf{\Lambda_{\epsilon}C\Lambda_{\epsilon}^{\dagger}h}$, where $\mathbf{y}$ and $\mathbf{h}$ are $n$-length complex-vectors, $\mathbf{C}$ is a circulant $n\...
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Exponential of a special matrix

Problem definition Given an integer $N>1$, let $A_N$ be the following $N\times N$ matrix \begin{equation*}A_N\triangleq \left[\begin{array}{c|c} & I_{N-1} \\ \hline 0_1 & \end{array}\...
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Absorbing generator matrix in continuous-time Markov chain models

Setting Let $[0,T]$ with $T\in\mathbb{R}^{+}$ be a time horizon over which $N\in\mathbb{N}^{+}$ continuous-time time-homogeneous Markov chains make transitions between $\{1,...,h\}$ states with $h\in\...
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Help understand the equality of variable substation.

Find general expression of $$\left[ \begin{matrix} \alpha & \beta \\ -\beta & \alpha \end{matrix} \right]^k.$$ We let $A$ the given matrix without the "$k$" power. And write $\alpha ...
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Solve a differential equation involving matrices.

Let $A = \left[ \begin{matrix} -1 & 1 \\ 0 & -1 \end{matrix} \right]$, $b = \left[ \begin{matrix} 2 \\ 3 \end{matrix} \right]$. Solve the differential equation $$\dot x = Ax + b; \quad x(0) = ...
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2 answers
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Let $A = \left[ \begin{matrix} I & X \\ 0 & -I \end{matrix} \right]$, $X\in\mathbb R^{m\times n}$ arbitrary. Need to show value of $e^A$.

Let $A = \left[ \begin{matrix} I & X \\ 0 & -I \end{matrix} \right]$, $X\in\mathbb R^{m\times n}$ arbitrary. Need to show value of $e^A = \left[ \begin{matrix} eI & \sinh 1X \\ 0 & \...
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Rigorously justifying Dirac representation

Dirac's equation necessitates the introduction of $4\times4$ complex matrices $\gamma^\mu$ ($\mu = 0, 1, 2, 3$) satisfying the Clifford algebra $$\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu \nu}$$ where $\...
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Closed form for 2 by 2 matrix exponential

I am trying to find the closed form solution for $e^{A}$, where $$A = \begin{pmatrix} \lambda & \mu_1 \\ \mu_2 & \lambda\end{pmatrix}. \quad \mu_1\mu_2 < 0$$ Is there a way to approach this ...
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-1 votes
1 answer
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Creating Exponential Matrices [closed]

Show thatI + (e^ t − 1)B$. I'm not very sure how to even start on this question.
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Integrate matrix exponential surrounded by vectors

Suppose $\boldsymbol{M}$ is an $n \times n$ non-singular matrix, $\boldsymbol{\beta}$ is a $1 \times n$ row vector, $\boldsymbol{1}$ is a $n \times 1$ column vector of ones. How do I compute the ...
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Show that matrices commute: if Exponential of linear combination of matrices is product of exponentials of the matrices

In this question it is discussed how the exponential of a linear combination of two matrices reduces to the product of the exponentials of the two matrices, given that they commute. $$exp(aX+bY)=exp(...
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