# Questions tagged [matrix-exponential]

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

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### 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-...
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### 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 ...
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### 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^{-...
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### 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}...
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### 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 ...
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### 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), ...
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### 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?
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### 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 ...
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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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### 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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### 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 ...
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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 ...
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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 ...
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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 \...