Skip to main content

Questions tagged [matrix-exponential]

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

Filter by
Sorted by
Tagged with
4 votes
1 answer
48 views

Rank of the matrix made by the diagonal components of the exponential matrix of the block diagonal matrix

I define the $9 \times 9$ matrix $\bf{K}$ as $${\bf K}=\exp\left(\begin{bmatrix} \bf{A} & p\bf{I} & \bf{O} \\ p\bf{I} & \bf{B} & q\bf{I} \\ \bf{O} & q\bf{I} & \bf{C} \\ \end{...
Sakurai.JJ's user avatar
2 votes
2 answers
141 views

Why the map $e^{A+B} = e^{A}e^B$ if $A,B$ are matrices that commute [duplicate]

Let $A,B$ be matrices with dimension $N$. Define $ e^A:= I+A+\frac{A^2}{2!}+.... = \lim_{n \to\infty} \sum_{k=0}^n\frac{A^k}{k!}.$ Prove using limits if $AB = BA$ then $ e^{A+B} = e^Ae^B.$ I have ...
mathnoob's user avatar
2 votes
0 answers
31 views

Bounding norms of symplectic matrix factorisations and non-seperable Hamiltonian flows

Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc} A & C\\ C^T & B\\ \end{array}\right)$ symmetric positive ...
Ben94's user avatar
  • 108
1 vote
1 answer
52 views

The rank of a matrix made by the diagonal components of the matrix exponential

I define the $4 \times 4$ matrix $K$ as $$K=\exp\left(\begin{bmatrix} \log{p} & a & 0 & 0 \\ a & \log{q} & 0 & 0 \\ 0 & 0 & \log{cp} & a \\ 0 & 0 & a & ...
Sakurai.JJ's user avatar
1 vote
1 answer
46 views

Operator exponential equality question: Does $X(\sigma) = Y\sigma$ imply $\exp(X)(\sigma) = \exp(Y)\sigma$?

See this question and other linked questions I am exploring on physics stack exchange. https://physics.stackexchange.com/questions/819663/ad-circ-exp-exp-circ-ad-and-ei-theta-2-hatn-cdot-sigma-sigma-e-...
Jagerber48's user avatar
  • 1,451
2 votes
1 answer
58 views

Representation/factorising of symplectic groups elements

According to Hall Chap. 3 Corollary 3.47: for a connected matrix Lie group $G$, every element $A\in G$ can be written in the form $A=e^{X_1}e^{X_2}...e^{X_k}$ for some $X_i\in g$, where $g$ is the Lie ...
Ben94's user avatar
  • 108
0 votes
1 answer
29 views

Adjoint with exponential map $e^{-A(t)} \left(\frac{d}{dt} \exp[B(t)] \right) e^{A(t)} =e^{-ad_A} \frac{d \exp[B(t)]}{dt}$

For a matrix, I know \begin{align} A e^{B} A^{-1} = e^{ABA^{-1}} = e^{Ad_A(B)} \end{align} Using the formula from Lie algebras, we have \begin{align} \frac{d}{dt} \exp[B(t)] = \exp[B(t)] \frac{1-e^{-...
phy_math's user avatar
  • 6,490
0 votes
1 answer
21 views

computaton of vector fields in KAK decomposition

I want to derive the following the left-invariant vector fields. \begin{align} {D}_t h(t) = e^{-ad \phantom{1} k_2(t)} e^{-ad \phantom{1} a(t)} {D}_t k_1(t) + e^{-ad \phantom{1} k_2(t)}{D}_t a(t) + {D}...
phy_math's user avatar
  • 6,490
0 votes
0 answers
24 views

Periodicity of matrix exponentials

Let $x \in \mathbb{R}^n$ and suppose we are trying to compute an integral of the form $$ I = \int d^n x \, f(x). $$ Here, I am interested in performing a change of variable $x \rightarrow \exp\left(i \...
Latos's user avatar
  • 37
1 vote
0 answers
22 views

Norm of the Exponential of a Scalar Multiple of a Matrix

Basically, the title. Suppose, we have some matrix (or operator) $A$ - can we relate $\left\lVert e^{A}\right\rVert_2$ to $\left\lVert e^{k A}\right\rVert_2$, where $k\in\mathbb{R}$? More generally, ...
gettingmathy's user avatar
1 vote
0 answers
98 views

Interpolation in $O(p,q,r,\mathbb{R})$

Definite Setting: $SO(n,\mathbb{R})$ vs $O(n,\mathbb{R})$ If I have a rotation matrix $R_0\in SO(n,\mathbb{R})$ and a rotation matrix $R_1 \in SO(n,\mathbb{R})$ I can interpolate between the two by ...
lightxbulb's user avatar
  • 2,109
1 vote
1 answer
31 views

Induced map on Lie algebra of product of Lie group morphisms

I am working with Lie groups and Lie algebras and have some trouble with proving something that I think is right. Let $G$ be a (simply connected) Lie group. Let $\mathfrak g = T_e G$ be its associated ...
noparadise's user avatar
2 votes
1 answer
72 views

Power series of matrices that is similar to hyperbolic sine

Consider a power series $$ \phi = \sum_{k=0}^{\infty} \frac{(M_2M_1)^kM_2}{(2k+1)!} $$ where $M_1$ and $M_2$ are symmetric positive semidefinite matrices, thus standard square roots $M_1^{\frac{1}{2}}$...
Eric J's user avatar
  • 21
1 vote
1 answer
37 views

Why ins't this SU(4) Matrix produced by the exponential map?

I was working with the SU(4) Lie group, which is compact and simply connected. This should imply that the exponential map is sujective on the group. However i came across the matrix $$G=\begin{pmatrix}...
Arthur's user avatar
  • 35
0 votes
0 answers
42 views

Inverse matrix elements from exponential integral

Based on the fact that: $$ \int_0^\infty dx\ e^{-x a} = \frac{1}{a}, \ \Re(a)>0, $$ and that: $$ \int_0^\infty dx\ e^{-x A} = A^{-1}, $$ if $A$ is an invertible matrix and $$ s(A):=\sup\{\Re(\...
Marcosko's user avatar
  • 175
3 votes
1 answer
74 views

What is in the image of the exponential of $\frak{sl}(n,\mathbb{R}$)? What do you need to get all of $\mathrm{SL}(n,\mathbb{R}$)?

This question discusses how $\mathrm{S}L(2,\mathbb{R}$) coincides with $\pm\exp(z)$ with $z\in \frak{sl}(n,\mathbb{R}$) (the real traceless matrices). Is it known what happens for $n>2$? Namely, ...
Another User's user avatar
3 votes
1 answer
145 views

Commutation relation between exponentials of Pauli matrices

Define $P_\phi := e^{-iP\phi}$, where $P$ is a Pauli matrix with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed (see Page 1 of this paper) that if $P'P = -PP'$ i.e. we have two ...
user1936752's user avatar
  • 1,708
2 votes
1 answer
89 views

Writing $\exp{tA}$ as finite sum

Let $A \in \mathcal{M}_{n\times n} (\mathbb{R})$ such that $A^2=\alpha A$ for some $\alpha \neq 0$. Under this assumption, we have by induction that $A^{n}=\alpha^{n-1}A$; $n \geq2$; then: $$\exp(tA)=...
J P's user avatar
  • 893
0 votes
1 answer
41 views

Exponential of a Multivector in Geometric Algebra: $\exp (xe_1 + ye_2 + be_1\wedge e_2)$

I'm working on understanding the exponential function applied to multivectors in the context of Geometric Algebra, specifically for the multivector $xe_1 + ye_2 + be_1\wedge e_2$. I have found ...
Anon21's user avatar
  • 2,589
0 votes
0 answers
37 views

Matrix Exponential and Conjugation of Jordan Normal Form and Real Normal Form

Given a diagonalizable matrix $A \in \mathbb{R^{N \times N}}$, we can decompose $A$ in following terms: $A = V \Lambda_{C} V^{-1}$ where $V, \Lambda_{C} \in \mathbb{C^{N \times N}}$ $A = Q \Lambda_{R}...
lostintimespace's user avatar
5 votes
0 answers
90 views

When does A and exp(B) commuting imply A commutes with B?

Let $A,B$ $\in GL_{n}(\mathbb{C})$ and $[A,B] = AB-BA = 0$. My question is about the existence of a $b \in M_{n}(\mathbb{C})$ such that $B = \exp(b) $ and $[A,b] =0 $. Note that in general $[A,\exp(b)...
arczn's user avatar
  • 51
1 vote
2 answers
115 views

Element-to-Element Expression of a Matrix Exponential?

Given a matrix $B = exp(A)$, how do we express $B_{ml}$ using $A_{ij}$? If $A$ is a diagonal matrix, this is very straightforward: $B_{ii} = exp(A_{ii})$. Can we generalize it to a real square matrix $...
lostintimespace's user avatar
0 votes
0 answers
126 views

Backpropagation: Chain Rule for Matrix Exponential?

Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a ...
lostintimespace's user avatar
1 vote
1 answer
42 views

Exponential identity involving differential operators

I stumbled upon an equation that could be an identity, but I'm not sure. $$ e^{a(\partial_x + f'(x))} e^{-a \partial_x} = e^{f(x+a) - f(x)} $$ The operator exponential can be understood as a power ...
Bio's user avatar
  • 1,066
0 votes
0 answers
23 views

Diagonalizing Block Matrices of Hilbert Space Operators

I’m running into difficulty computing operator exponentials for “block matrices” of Hilbert space operators. It would be extremely useful to be able to diagonalize these block matrices. While the ...
Joe's user avatar
  • 2,968
0 votes
0 answers
30 views

Matrix exponential of nonsingular matrix, diagonalizable?

Let $A\in \mathbb{R}^{n \times n}$, and be nonsingular. Is $e^A$ (matrix exponential) diagonalizable? Can you prove it? If not correct, what are the conditions for $A$ so that $e^A$ is diagonalizable. ...
Mathisfreedom's user avatar
1 vote
0 answers
44 views

Matrix Exponential for Rotations [closed]

Are the matrix exponentials of real skew-symmetric matrices always rotations? Does that mean that when we have any real skew-symmetric matrix in n-Dimensions, we can form a rotation around the ...
user avatar
0 votes
1 answer
53 views

A problem on matrix exponential [closed]

I know that $$e^{M}=\sum_{k=0}^{+\infty} \frac{M^k}{k!}$$ But from this I stuck
urt43as's user avatar
  • 341
0 votes
3 answers
87 views

Skew-symmetric matrices and the fact that their exponential matrix is orthogonal

I want to ask why "If $A$ is skew-symmetric ($A^T=-A$) then $e^{At}$ is an orthogonal matrix". Here is my solution step: $e^{At}*(e^{At})^T =e^{At}*e^{-At}=e^{At-At}=e^0$ I think the matrix $...
Ruizhe Pang's user avatar
0 votes
0 answers
58 views

What are the properties of the domain and image of the matrix exponential $e^A$?

I know that for real numbers $\exp(x)$ is defined for every real $x$ and that the result is always greater than zero. I want to compare that to the case $\exp(A)$ where $A$ is a square matrix. Is it ...
Luiz Phillyp Sabadini Bazoni's user avatar
1 vote
1 answer
53 views

Matrix exponential in terms of submatrices

I want to compute the exponential of a matrix: $$ A = \begin{bmatrix} A_{11} & I_{3\times3} & \boldsymbol{0}_{3\times3}\\ \boldsymbol{0}_{3\times3} & \boldsymbol{0}_{3\times3} & I_{3\...
Ogiad's user avatar
  • 133
0 votes
1 answer
34 views

IMU preintegration: rotation increment derivation

I am reading article on IMU preintegration: https://rpg.ifi.uzh.ch/docs/TRO16_forster.pdf. I am not too strong in math and I fail to understand how some derivations were created. One of such equations ...
aquila's user avatar
  • 109
0 votes
0 answers
141 views

The matrix exponential of a positive definite matrix is also positive definite [duplicate]

Let $L$ be a $n \times n$ symmetric positive definite matrix. Prove that for $t>0$ that the matrix exponential $e^{-tL}$ is also symmetric positive definite.
Eto's user avatar
  • 57
0 votes
0 answers
32 views

Linear phase shifts in the exponential of special tridiagonal matrices

I've been working on a physics problem that has led me to make the following numerical observation. Let $L$ be an $N\times N$ real-symmetric tridiagonal matrix whose diagonal entries are zero. This ...
miggle's user avatar
  • 285
0 votes
1 answer
46 views

Truncated BCH expansion [closed]

Is it true that once a term in the Baker-Campbell-Hausdorff expansion is zero, then all the following terms must also be zero? For example, it is not immediately obvious to me that if $[Y, [X, [X, Y]]]...
Creeptographer's user avatar
0 votes
0 answers
70 views

Writing matrix in complex with mutiple parameters exponential form

I'm trying to take the continuum limit of a quantum walk, which involves writing the quantum 'coin' in exponential form. This is essentially just writing a matrix in exponential form. Most literature ...
Saz's user avatar
  • 33
0 votes
0 answers
57 views

Exponential map for matrix Lie groups and smallness of the argument

From the Wikipedia page of the Adjoint Representation we can read If $G$ is an immersed Lie subgroup of the general linear group $\mathrm {GL} _{n}(\mathbb {C} )$ (called immersely linear Lie group), ...
Gabriel Ybarra Marcaida's user avatar
0 votes
0 answers
66 views

Logarithm of identity matrix

What are the possible $n \times n$ matrices such that $e^X = I$? Some obvious ones are $X = 2\pi i m I$ for integer $m$. Are there others and do they have some kind of group structure?
Bio's user avatar
  • 1,066
0 votes
1 answer
61 views

Convergence time for system of linear ODEs

Given a real symmetric negative definite matrix $\mathbf{A}$, consider a system of linear ODEs given by $$ \dot{\mathbf{x}} = \mathbf{A} \left( \mathbf{x} - \mathbf{x}_\star \right),$$ whose solution ...
cisprague's user avatar
  • 169
1 vote
1 answer
119 views

How to calculate time ordered Exponential (special case of rotation matrix)

I have the following problem $$ \dot{R}(t)=R(t) \begin{pmatrix} 0 & 0 & -\cos{t} \\ 0 & 0 & -\sin{t} \\ \cos{t} & \sin t & 0 \end{pmatrix} \equiv R(t) A(t), \quad R(0)=\mathbb{...
Motoko's user avatar
  • 103
1 vote
1 answer
92 views

Does $(e^a)^b=e^{ab}$ hold for the matrix exponential?

Here's the textbook definition of the matrix exponential. Let $A$ be a $n\times n$ matrix, then $$e^A=\sum_{n=0}^{\infty} \dfrac{1}{n!}A^{n}.$$ It's not quite clear to me how to work this series out. ...
vshp11's user avatar
  • 53
1 vote
2 answers
142 views

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 & ...
Aner's user avatar
  • 320
1 vote
2 answers
85 views

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 ...
adrimsvieira's user avatar
0 votes
0 answers
38 views

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 ...
adrimsvieira's user avatar
0 votes
0 answers
28 views

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: ...
Avraham's user avatar
  • 91
0 votes
0 answers
30 views

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 ...
iago-lito's user avatar
  • 539
0 votes
1 answer
80 views

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 ...
J.Agusti's user avatar
  • 155
1 vote
0 answers
72 views

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 ...
user0134's user avatar
  • 404
2 votes
0 answers
80 views

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 ...
user160623's user avatar
2 votes
0 answers
17 views

$n$-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 \...
user82261's user avatar
  • 1,257

1
2 3 4 5
15