Questions tagged [matrix-exponential]
"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."
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Integrate product of matrix exponentials of a symmetric matrix
Let $\mathbf{A}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible, symmetric matrix.
Let $\mathbf{Q}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible matrix.
Given these properties of $\mathbf{...
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For a matrix $A(z)$ that represents the operation of multiplication with a complex number $z$, what does it mean for $e^{A(z)t} = A(e^{zt})$?
We can have a complex number $z = a + bi$ that determines a matrix $A(z)$ in the following way:
$$ A(a + bi) = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $$
This matrix represents the ...
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Exponeintal of symmetric triangular matrix
I want to know the exponeintal of given $n \times n$ symmetirc real tridiagonal matrix ${\bf K}_n$, which is defined as
$${\bf K}_n=\begin{bmatrix}
0 & a & 0 & 0 & \dots & 0 & ...
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Rank of the matrix made by the diagonal components of the exponential matrix of the block diagonal matrix
I define the $9 \times 9$ matrix $\bf{K}$ as
$${\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{...
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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 ...
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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 ...
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The rank of a matrix made by the diagonal components of the matrix exponential
I define the $4 \times 4$ matrix $K$ as
$$K=\exp\left(\begin{bmatrix}
\log{p} & a & 0 & 0 \\
a & \log{q} & 0 & 0 \\
0 & 0 & \log{cp} & a \\
0 & 0 & a & ...
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Operator exponential equality question: Does $X(\sigma) = Y\sigma$ imply $\exp(X)(\sigma) = \exp(Y)\sigma$?
See this question and other linked questions I am exploring on physics stack exchange. https://physics.stackexchange.com/questions/819663/ad-circ-exp-exp-circ-ad-and-ei-theta-2-hatn-cdot-sigma-sigma-e-...
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Representation/factorising of symplectic groups elements
According to Hall Chap. 3 Corollary 3.47: for a connected matrix Lie group $G$, every element $A\in G$ can be written in the form $A=e^{X_1}e^{X_2}...e^{X_k}$
for some $X_i\in g$, where $g$ is the Lie ...
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Adjoint with exponential map $e^{-A(t)} \left(\frac{d}{dt} \exp[B(t)] \right) e^{A(t)} =e^{-ad_A} \frac{d \exp[B(t)]}{dt}$
For a matrix, I know
\begin{align}
A e^{B} A^{-1} = e^{ABA^{-1}} = e^{Ad_A(B)}
\end{align}
Using the formula from Lie algebras, we have
\begin{align}
\frac{d}{dt} \exp[B(t)] = \exp[B(t)] \frac{1-e^{-...
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computaton of vector fields in KAK decomposition
I want to derive the following the left-invariant vector fields.
\begin{align}
{D}_t h(t) = e^{-ad \phantom{1} k_2(t)} e^{-ad \phantom{1} a(t)} {D}_t k_1(t) + e^{-ad \phantom{1} k_2(t)}{D}_t a(t) + {D}...
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Periodicity of matrix exponentials
Let $x \in \mathbb{R}^n$ and suppose we are trying to compute an integral of the form
$$ I = \int d^n x \, f(x). $$
Here, I am interested in performing a change of variable $x \rightarrow \exp\left(i \...
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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, ...
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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 ...
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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 ...
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