14
votes
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
Counterexample: Consider the entire function $$ A(z) = \pmatrix{e^z & 0\cr
z & 1\cr}$$
An entire logarithm of $A(z)$ must have eigenvalues $z + 2\pi i n$ and $2 \pi ...
- 54.2k
14
votes
Accepted
exponential/logarithm for unipotent algebraic groups
This is false in characteristic $p$, no matter how large $p$ is. The counterexample is the group parameterized by
$\begin{pmatrix} 1 & t & t^p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{...
- 148k
13
votes
Accepted
Is the set of real matrices with at least one real logarithm closed under multiplication?
This is already not true for $2$-by-$2$ matrices: Consider
$$
A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad
\text{and}\quad
B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}.
$$...
- 108k
12
votes
Matrix elements of exponential of tridiagonal matrices
Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of ...
- 20.2k
10
votes
Accepted
Integral of the entrywise square of the exponential of a matrix
Inspired strongly by Anthony's answer, here is a formula that works for arbitrary $A$. Let $M$ be the $n^2 \times n^2$ square matrix given by $$M= A \otimes I_n + I_n \otimes A_n$$
i.e. in terms of ...
- 148k
9
votes
Logarithm of a bounded operator
The other answer is not correct. If $L$ is the half-line issuing from the origin, then we can find a branch of the logarithm that is holomorphic on $\mathbb{C}\setminus L \supset{\rm spec}(A)$. Then ...
- 42.8k
6
votes
Accepted
Kronecker product: Is it possible to simplify this product $e^{-A} \otimes e^{A}$ where $A$ is an invertible and symmetric matrix
One can diagonalize your $A = VDV^{-1}$ explicitly; the closed formulas are here for instance.
Once you have those matrices, you can write the orthogonal eigendecomposition
$$
\exp(-A) \otimes \exp(A) ...
- 20.2k
6
votes
exponential/logarithm for unipotent algebraic groups
EDIT: I roll back to the previous proof, in characteristic 0 only. My last proof including characteristic $p$ was false (thanks to Will Sawin for noticing this).
Let $G={\rm GL}_{n,k}$\,, where $k$ ...
- 14.1k
5
votes
Accepted
Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
This Asymptote code seems to work perfectly and for any $t$ in your range the estimate uses $Cd$ operations and is a guaranteed upper bound though I am not sure whether $C$ is small enough for you (I ...
- 61.9k
5
votes
Accepted
Attempts to define a matrix exponential over (as much as possible) general fields
You can give an equivalent definition of the matrix exponential. or any matrix function $f$, using the Jordan form. If $A = VJV^{-1}$, and $J$ is the direct sum of diagonal blocks of the form
$$
J_i = ...
- 20.2k
5
votes
Property for bounding matrix exponential
For any $n\times n$ matrices $X_1$ and $X_2$,
\begin{equation}
\begin{aligned}
e^{X_1+X_2}-e^{X_1}&=\sum_{k=0}^\infty\frac1{k!}\,[(X_1+X_2)^k-X_1^k] \\
&=\sum_{k=0}^\infty\frac1{k!}\;
...
- 128k
4
votes
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
This is my comment above slightly expanded. Let's focus on $2\times 2$ matrices for convenience and let $A(z)$ be entire with $\det A=1$ (divide through by a holomorphic square root of $\det A$ if ...
- 24.2k
4
votes
Inequality with matrix exponentials. Is it true that $x^\top e^{-xx^\top - AA^\top} x \leq x^\top e^{-xx^\top} x$?
Alas, this is not correct. You may try $d=2$, $x=(a, a)^\top$ for certain $a>0$, and choose $A$ of rank 1 such that $AA^\top+xx^\top=\operatorname{diag}(p, q)$. For positive $p,q$ such matrix $A$ ...
- 109k
3
votes
Integral of the entrywise square of the exponential of a matrix
You can do something. Here's a computation for diagonalizable $A$. Let $A=BDB^{-1}$ and let the elements of $D$ be
$-\lambda_1,\ldots,-\lambda_d$. Then
\begin{align*}
\int_0^\infty (e^{At})_{ij}^2\,...
- 23.2k
3
votes
Properties of matrix exponential without using Jordan normal forms
I do not know if this is what you are after. The following shows that the Jordan form can be replaced by the Schur form. This feels a bit like cheating. Is this enough? The trickier parts are the ...
- 1,167
3
votes
Accepted
Behavior of a Baker-Campbell-Hausdorff problem at infinity
It may be helpful to consider an example that can be solved exactly; Using the Special-case closed form of the Baker-Campbell-Hausdorff formula one finds that if the commutator $[X,Y]$ evaluates to
$$[...
- 188k
3
votes
Property for bounding matrix exponential
The wikipedia article you refer to is here. This indeed states that
$$
\|e^{X+Y}-e^X\| \le \|Y\| e^{\|X\|}e^{\|Y\|} \tag{1}\label{1}
$$
for any "matrix norm", and, according to the link that ...
- 24.2k
2
votes
Logarithm of a bounded operator
There is a Banach algebra version of the result (including the Banach space version) in Theorem 10.30 (page 264) of W. Rudin's book Functional analysis, 2nd ed. McGraw-Hill 1991.
- 4,461
2
votes
Accepted
Higher order Lyapunov equation
Inspired by Will's answer of Integral of the entrywise square of the exponential of a matrix I realized that the above equation can also be computed by using the Kronecker product.
Let $$M= A \oplus ...
- 562
2
votes
Properties of matrix exponential without using Jordan normal forms
You can use a weaker version of Jordan normal form, namely an upper triangularization. There is a very straightforward conceptual proof that every square matrix over an algebraically closed field $k$ ...
- 118k
2
votes
Accepted
Reorganizing the terms in the Baker–Campbell–Hausdorff formula (or Zassenhaus formula) for $\exp(X+\delta Y)$ for small $\delta$
This expansion is derived by K. Kumar in On Expanding the Exponential, see equation (9) (with $t=1$) and section 6.
- 188k
2
votes
Accepted
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
In terms of Pauli matrices:
$$U=u_1I+iu_2\sigma_3+iu_3\sigma_2+iu_4\sigma_1,\;\;u_1^2+u_2^2+u_3^2+u_4^2=1,$$
$$V=\alpha (n_1\sigma_1+n_2\sigma_2+n_3\sigma_3),\;\;n_1^2+n_2^2+n_3^2=1,$$
$$\exp(iV)=I\...
- 188k
2
votes
Limiting value of expectation of trace of exponential of Wishart matrix
A closed-form expression is not forthcoming, but here are the plots of $\mathbb{E}[\text{trace}\, T]$ (left plot) and $\mathbb{E}[\text{trace}\, TS]$ (right plot) as a function of $\gamma$ in the ...
- 188k
2
votes
Matrix logarithm for d-dimensional cyclic permutation matrix
The article below presents the general form:
https://arxiv.org/abs/2001.11909
- 21
1
vote
exponential/logarithm for unipotent algebraic groups
In characteristic $p>0$ you can expect a property of this kind in nice situations with certain precautions.
First precaution is that a Lie subalgebra ${\mathfrak g} \leq Lie (GL_n)$ may be tangent ...
- 12.3k
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