Questions tagged [matrix-exponential]

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

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35 views

What is the length of an Exponential map?

The exponential map for the unit circle is $it \to \exp (it) = \cos t + i \sin t$ This group can be augmented with a norm on the complex plane such that $r$ is the length of the complex number on the ...
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1answer
54 views

When a $A$ commute with $e^B$, does it also commute $B$?

Let $A$ and $B$ be $n\times n$ matrices, and let $[A,B]=AB-BA$ be the commutator of $A$ and $B$. Then, if we consider the matrix exponential $e^B=\sum_n \frac{B^n}{n!} $, we have $$ [A,e^B]=\sum_n \...
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1answer
31 views

Finding a vector subspace

Considering the exponential map $\exp : M_{n*n}(\mathbb{C}) → GL(n, \mathbb{C})$ we know that it is a local diffeomorphism at $0$. How can we find the appropriate vector subspaces $V( U(n))$ such that ...
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1answer
34 views

Exponential of upper triangular matrix

Consider the matrix $$ A=\left(\begin{matrix}a&b\\0&a\end{matrix}\right),\qquad A^n=\left(\begin{matrix}a^n&b(na^{n-1})\\0&a^n\end{matrix}\right) $$ The exponential series is $$ \exp(...
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14 views

Length of a vector squared under the exponential map, correct formula and properties thereof?

Let's say we're working in $\mathbb{R}^{n}$ with a set of coordinates $\phi^{1},\phi^{2}...\phi^{n}$. I'm looking at the length squared of some arbitrary vector given by the coordinates: $$|\phi|^{2}=\...
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37 views

Matrix exponential of a Kronecker product

I'm trying to find an expression for the matrix exponential of a Kronecker product of two matrices, $\hat{c}$ and $\hat{D}$. The matrix $\hat{c}$ is a small real and symmetric $2\times 2$ matrix: $$ \...
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1answer
49 views

conjugate function of log trace of matrix exponential

Given an n-by-n real symmetric matrix X, define $$e^X = \sum_{k=0}^{n}\frac{X^k}{k!}$$ Derive the Fenchel conjugate of $$f(X) = log(Tr(e^X))$$ as a function on n-by-n real symmetric matrices. Here Tr ...
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1answer
48 views

find the matrix exponential

Let \begin{equation*} A = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}\end{equation*} \begin{equation*} B = \begin{pmatrix} 2 & 1 & 0\\ 0 & 2 &...
3
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1answer
76 views

Find function f(z) for a matrix

Let $z \in \mathbb{C}$. Let $ f(z)= \begin{cases} 1,& \text{if } |z|<\frac{1}{2}\\ 0, & \text{if} |z-2| <\frac{1}{2} \\ 0, & \text{if} |z-3| <\frac{1}{2} \end{...
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1answer
80 views

find the exponential form of a matrix [closed]

Let \begin{equation*} A = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}\end{equation*} \begin{equation*} B = \begin{pmatrix} 2 & 1 & 0\\ 0 & 2 &...
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1answer
50 views

Is there an efficient systematic way to find relations between powers of Matrices?

Hi I have a 3 by 3 matrix of the following form: $$ A=\begin{bmatrix} a & b & 0 \\ -b & a & c \\ 0 & -c & d \\ \end{bmatrix} $$ $$a,b,c,d\in\Re $$ I'm trying to explore ...
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1answer
47 views

Let $V$ be the vector space polynomials of degree less than $n$ over $\mathbb{R}$

Let $V$ be the vector space polynomials of degree less than $n$ over $\mathbb{R}$. In other words $$V = \{p\in R[x] \mid \deg p < n\}.$$ Let $T : V \rightarrow V$ be the map $T =\frac{d}{dx}$. From ...
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1answer
28 views

Determinant of identity matrix minus exponential matrix $\det(I-k \exp (M)),$

I am currently struggling with the following determinant $$\det(I-k \exp (M)),$$ where $I$ is the $2\times2$ identity matrix, $M$ is a $2\times2$ matrix and $k$ is an arbitrary constant. Is there a ...
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3answers
75 views

$ A=\begin{pmatrix} 0 & 2 & 0\\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{pmatrix}$, calculate $e^A$

I have encounter a question in my book , it was For $ A=\begin{pmatrix} 0 & 2 & 0\\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{pmatrix}$, calculate $e^A$ My solution way : I tried to find its ...
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0answers
33 views

Differential equation solution to initial value problem

I recently went across the IVP and decided to explore it for my class. Though I still don't get how to get the solution. I have worked some of it up but I'm stuck, does anyone know how to continue? ...
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1answer
43 views

The measure of the image of the exponential of real matrices

It is known that the matrix exponential over the real matrices $\exp : M_n(\mathbb{R}) \to GL_n(\mathbb{R})$ is not surjective and that its image $S $ is the subset of all invertible matrices that are ...
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1answer
44 views

Exponential parametrization for diagonal SL(2,R)

I already saw this question (and the linked questions) but mine is slightly different. I am asked to prove the fact that the exponential parameterization isn't a good one for $SL(2,\mathbb{R})$ using ...
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24 views

Fudamental matrix of the adjoint system of homogeneous differential equation

Prove that if $y'(t)=A(t)y(t)$ and $\psi(t)=e^{At}$ then $\eta(t)$ is the fudamental matrix of the adjoint system iff $\psi^*\eta$ is a constant invertible matrix. $A^*(t)$ is the conjugate transpose ...
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1answer
85 views

Solution for rotation $d\vec{r}/dt = \vec{\omega}\times\vec{r}$.

Let's consider an equation: \begin{equation} \frac{d\vec{r}}{dt}=\vec{\omega}\times\vec{r}. \end{equation} This equation might be expressed via "angular velocity" matrix: \begin{...
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Applications of the weighted matrix exponential $x^\top e^A x$ (a quadratic function)

In this post, the matrix exponential $$e^A = \sum_{k=0}^\infty \frac{1}{k!} A^k$$ of any symmetric matrix $A$ was proven to be positive-definite based on the quadratic function $$x^\top e^A x$$ where $...
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1answer
41 views

Decomposition of the matrix exponential (of the covariance matrix)

The sample covariance matrix $\Sigma$ (chosen because it is symmetric and, therefore, diagonalizable) can be decomposed as $$\Sigma = \text{diag}(\sigma) C \text{diag}(\sigma)$$ where $\text{diag}(\...
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31 views

Compute exponential of sum of Kronecker products of non-commuting matrices

In the context of CTMCs I want to compute following matrix exponential: $$\exp(M_1\otimes M_2 + M_3\otimes M_4 + D_1\otimes D_2 + D_3\otimes D_4)$$ with: $\otimes$ being the Kronecker product, all ...
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1answer
44 views

Solving multi DOF damped system using block matrix and matrix exponential

I'm trying to solve a multi DOF damped system for $U$. $[M]$ and $[K]$ are symmetric nxn matrices, but $[C]$ is not symmetric. Can I use the following approach? Put in block matrix form Solve the 1st ...
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56 views

On a limiting value of a matrix exponential

$\mathbf {The \ Problem \ is}:$ If $A$ and $B$ are two bdd linear operators on a Banach space, then show that $\lim_{n \to \infty} (e^\frac{A}{\sqrt n}e^\frac{B}{\sqrt n}e^\frac{-A}{\sqrt n}e^\frac{-B}...
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1answer
43 views

Riemannian exponential map on symmetric spaces of non-compact type

Let $S$ be a symmetric space of non-compact type, specifically let $S= SL(n)/SO(n)$ for some $n\geq 2$. I know that such spaces are geodesically complete, simply connected, and have non-positive ...
2
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1answer
67 views

Calculating series expansions within a matrix: matrix exponential

I have a $(3 \times 3)$ matrix $$ Y = \begin{pmatrix} 0 & - e^{-i \theta} & 0 \\ e^{i \theta} & 0 & - e^{-i \theta} \\ 0 & e^{i \theta} & 0 \end{pmatrix} $$ for which I would ...
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63 views

How to prove that the element-wise exponential of a symmetric matrix is not always positive-definite? [duplicate]

The matrix exponential $e^X$ of a square symmetric matrix $X$ is always positive-definite, where $$ e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k $$ Does the exponentiation of each element of the symmetric ...
2
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1answer
41 views

Computing limits in matrix exponential

If $A \in B(V)$ where $V$ is a Banach space, and $B(V)$ is set of all bounded linear operators on $V$, then show if $lim_{n→\infty} A_n=A$, then $lim_{n→\infty}(I+\frac{A_n}{n})^n = exp(A)$ I am very ...
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0answers
31 views

Find matrix-valued function with a given derivative

Given $t$ and matrix-valued function $\mathbf{X}_t$, is it possible to find (analytically or computationally) matrix-valued function $\mathbf{C}_t$ such that the following holds? $$\mathbf{X}_{t} = \...
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1answer
40 views

Proving the logarithm rule for the matrix exponential

I am currently doing some work out of Undergraduate Analysis by Serge Lang and I have run into a bit of trouble with this question on the matrix logarithm. I showed that on the space of $n\times n$ ...
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1answer
51 views

Matrix exponential of a skew-symmetric Toeplitz matrix

From today's exam: Given the following matrix, $$B = \begin{pmatrix} 0 & -2/3 & 1/3\\ 2/3 & 0 & -2/3\\ -1/3 & 2/3 & 0\end{pmatrix}$$ Prove that $$\exp(aB) = I + \sin(a) B + (1 ...
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1answer
72 views

Given a solution to a system of ODE, prove that A has some pure imaginary eigenvalue (non zero). [closed]

$A$ is an $n$-dimensional square matrix: Let $X(t)$ be the solution of the problem of initial value $\frac{d}{dt}X=AX$ with $X(t_0)=X_0≠0$. Suppose there exists a real $t_1$ such that $X(t_1)=-X_0$. ...
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4answers
289 views

Calculating matrix exponential

Given matrix $$M = \begin{pmatrix} 7i& -6-2i\\6-2i&-7i\end{pmatrix}$$ how do I calculate matrix exponential $e^M$? I know I can use that $e^A=Pe^DP^{-1}$ where $D=P^{-1}AP$. I computed the ...
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1answer
105 views

Computing the Integral of a Complicated Matrix Exponential

For each $i = 1,2,\cdots,m$, let $A_i \in \mathbb{R}^{m\times m}$ be full rank, and $v_i \in \mathbb{R}^m$. Let us use the notation $[v_i]_{i=1}^m$ for the $m\times m$ matrix created by stacking $v_i$ ...
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1answer
53 views

Proving matrix exponential

Can anyone tell me how the following is derived? where $A$ is a matrix.
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How to evaluate the exponential integrator $\varphi$ functions using the matrix exponential

The Problem When using exponential integrators as shown here, the $\varphi$ functions need to be evaluated in each stage of the method, which are as follows: $$ \varphi _{0}(z)=e^{z},\quad \varphi _{k}...
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2answers
81 views

The matrix function $\frac{e^x - 1}{x}$ for non-invertible matrices

I am trying to evaluate the matrix function $$ f(X) = \frac{e^X - I}{X}, \tag{1} $$ where $e^X$ is the usual matrix exponential, defined by $$ e^X = \sum_{n=0}^{\infty} \frac{1}{n!} X^n. $$ The ...
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1answer
66 views

Can every SU(4) matrix be written as a product of of these 4 matrices

Consider a general $SU(4)$ matrix $$U=e^{i a^j T^j},$$ where the generators $T^i$ are chosen in the basis $$\{T\} = 1_2 \times \sigma^i, \quad \sigma^i \times 1_2, \quad \sigma^i \times \sigma^j, $$ ...
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0answers
30 views

Taylor expansion for matrix exponential

Consider the matrix exponential $$U(x)=e^{ix^j T^j}$$ where $T^j$ are matrices (in my particular application $U\in SU(4)$ and $T^j$ are its generators) and $x^j \in \mathbb{R}$. I would like to know ...
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3answers
113 views

Radius of convergence for matrix exponential

In the context of systems of linear ODE with constant coefficients, my lecture notes on ODE mention that the matrix exponential $e^{tA}$ has an infinite radius of convergence. This shows up in a proof ...
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0answers
51 views

Upper bound of the remainder term of $\int_0^t e^{A \tau} d\tau$ using Lagrange Remainder of the Taylor series

$\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$Let $A$ be an $n\times n$ real matrix which has real block diagonal form and each block has a ...
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1answer
42 views

Matrix exponential when $(Av.v) \leq -(v.v)$

I have a question from a previous homework that I never ended up figuring out: If $(Av.v) \leq -(v.v)$ for all $v \in V$ (where $V$ is finite dimensional vector space with an inner product), show that ...
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0answers
34 views

Matrix exponential: Prove that $\mathrm{e}^A=\lim_{N\rightarrow\infty}(I+\frac{A}{N})^N.$

Let $A\in\mathbb{C}^{n\times n}$. Define the matrix exponential $$ \mathrm{e}^A=\lim_{N\rightarrow\infty}\sum_{j=0}^N \dfrac{A^j}{j!}. $$ Prove that $$\mathrm{e}^A=\lim_{N\rightarrow\infty}(I+\frac{A}{...
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0answers
38 views

Derivative of exponential tensor with respect to a vector

Suppose you have a rank 4 Tensor $T$ whose coordinates are, in some basis, $T_{ijkl}$. Say the indices take values in some representation of $T$. Now suppose I construct a matrix by dotting this ...
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0answers
13 views

What are the intermediate steps of a canonical transformation of a Hamiltonian by an exponentiated matrix?

Background I have been reading through the Supplementary Notes from [1] and I am having some trouble understanding part of the derivation. I originally thought that this was a question for the Physics ...
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1answer
67 views

Is this a typo in Control Theory book?

I am currently reading a book on control theory and I have to calculate $e^{At}$ where $ A=\begin{pmatrix} 1 & 1 \\ 0 & -2 \end{pmatrix}$ I computed : $e^{At}=\begin{pmatrix} e^t & \frac{...
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0answers
32 views

Confusion over the matrix-exponential method

Suppose we have this matrix differential equation: $$ \vec x '= A \vec x $$ then we can use the matrix-exponential method to find a solution, given by: $$ \vec x = e^{At} \vec c $$ with $c$ the column ...
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2answers
114 views

Exponential operator expansion

In my lectures, the professor discussed that for exponential linear operators it is $$ \exp(\lambda A + \lambda B) \neq \exp(\lambda A)\exp(\lambda B) $$ for $AB\neq BA$. Now I know that the ...
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0answers
26 views

Proving $‖\exp(Ax)-e^{-\eta x}\sum_{n=0}^l\frac{(\eta x)^n}{n!}P^n‖<\epsilon$

I'm having trouble knowing where to start with this assignment. Let $A = \{a_{ij}\}_{i,j=1,\dots,n}$ be a matrix, and let $\eta = max_i(-a_{ii})>0$. With $P=I+\frac{1}{\eta}A$, where $I$ is the ...
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0answers
27 views

Confusion regarding $e^{I}$ and $e^{A \otimes I}$ and $e^{A \otimes B}$

I am a little confused regarding how the differences between $e^{I}, e^{A \otimes I}$ and $e^{A \otimes B}$ emerge in regards to how the taylor series acts on them. $$\displaystyle e^I = \sum_{k=0}^\...

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