Intereting Posts

If $x^4 \equiv -1 \mod p$ then $p \equiv 1 \mod 8$
Are sets constructed using only ZF measurable using ZFC?
Integrate: $\int_0^{\pi} \log ( 1 – 2 r \cos \theta + r^2)d\theta$
Integral of $1/(x^2-1)^2$
Given $BA$, find $AB$.
Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?
A Mathematical Coincidence, or more?
Archimedean property
Is $\log(z^2)=2\log(z)$ if $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$?
Does there exist any surjective group homomrophism from $(\mathbb R^* , .)$ onto $(\mathbb Q^* , .)$?
Upper and lower integration inequality
Prove if $f(a)<g(a)$ and $f(b)>g(b)$, then there exists $c$ such that $g(c)=f(c)$.
Convergence in law implies uniform convergence of cdf's
Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix:

I am working on the following problem. Let $e^{Mt} = \sum\limits_{k=0}^{\infty} \frac{M^k t^k}{k!}$ where $M$ is an $n\times n$ matrix. Now prove that

$$e^{(M+N)} = e^{M}e^N$$

given that $MN=NM$, ie $M$ and $N$ commute.

Now the left hand side of the desired equality is

$$e^{(M+N)} = I+ (M+N) + \frac{(M+N)^2}{2!} + \frac{(M+N)^3}{3!} + \ldots $$

On the right hand side of the equation we have

$$e^Me^N = \left(I + M + \frac{M^2}{2!} + \frac{M^3}{3!}\ldots\right) \left(I + N + \frac{N^2}{2!} + \frac{N^3}{3!} \ldots\right) $$

Now basically this is as far as I got… I am unsure on how to work out the product of the two infinite sums. Possibly I need to expand the powers on the left hand side expression but I am unsure how to do this in an infinite sum… If anyone could give me an answer or a hint that can help me forward I would greatly appreciate it. Thanks

- Each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$?
- Rings with isomorphic proper subrings
- How to show that the nth power of a $nxn$ nilpotent matrix equals to zero $A^n=0$
- A problem on condition $\det(A+B)=\det(A)+\det(B)$
- Find the number of $n$ by $n$ matrices conjugate to a diagonal matrix
- How to diagonalize a large sparse symmetric matrix to get the eigenvalues and eigenvectors

- $2\times2$ matrices are not big enough
- Fredholm alternative theorem for matrices
- What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?
- Hermitian matrix such that $4M^5+2M^3+M=7I_n$
- Show that for an invertible matrix, the images of a set of vectors spanning the space also form a spanning set
- How to divide polynomial matrices
- If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible
- What operations can I do to simplify calculations of determinant?
- to prove $f(P^{-1}AP)=P^{-1}f(A)P$ for an $n\times{n}$ square matrix?
- Extending a Chebyshev-polynomial determinant identity

Another take on it, which avoids the somewhat tedious term-by-term manipulation and term-by-term comparison of matrix power series:

Consider the ordinary, constant coefficient, matrix differential equation

$dX / dt = (M + N)X; \, \text{with} \, X(0) = I; \tag{1}$

the unique matrix solution is well-known to be

$X(t) = e^{(M + N)t}. \tag{2}$

Next, set

$Y(t) = e^{Mt}e^{Nt} \tag{3}$

and note that, by the Leibnitz rule for derivatives of products,

$dY / dt = (d(e^{Mt}) / dt) e^{Nt} + e^{Mt}(d(e^{Nt}) /dt) = Me^{Mt}e^{Nt} + e^{Mt}Ne^{Nt}, \tag{4}$

and since $MN – NM = [M, N] = 0$ we also have $[e^{Mt}, N] = 0$ so that (4) becomes

$dY / dt = Me^{Mt}e^{Nt} + Ne^{Mt}e^{Nt} = (M + N)e^{Mt}e^{Nt} = (M + N)Y, \tag{5}$

and evidently

$Y(0) = I, \tag{6}$

so that $X(t)$ and $Yt)$ satisfy the same differential equation with the same initial conditions; thus $X(t) = Y(t)$ for all $t$, or

$e^{(M + N)t} = e^{Mt}e^{Nt} \tag{7}$

for all $t$. Taking $t = 1$ yields the requisite result.*QED*

Hope this helps. Cheerio,

and as always,

**Fiat Lux!!!**

$\newcommand{\+}{^{\dagger}}%

\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%

\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%

\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%

\newcommand{\dd}{{\rm d}}%

\newcommand{\isdiv}{\,\left.\right\vert\,}%

\newcommand{\ds}[1]{\displaystyle{#1}}%

\newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}%

\newcommand{\expo}[1]{\,{\rm e}^{#1}\,}%

\newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}%

\newcommand{\ic}{{\rm i}}%

\newcommand{\imp}{\Longrightarrow}%

\newcommand{\ket}[1]{\left\vert #1\right\rangle}%

\newcommand{\pars}[1]{\left( #1 \right)}%

\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}

\newcommand{\pp}{{\cal P}}%

\newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}%

\newcommand{\sech}{\,{\rm sech}}%

\newcommand{\sgn}{\,{\rm sgn}}%

\newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}

\newcommand{\ul}[1]{\underline{#1}}%

\newcommand{\verts}[1]{\left\vert #1 \right\vert}%

\newcommand{\yy}{\Longleftrightarrow}$

\begin{align}

\color{#0000ff}{\large\expo{M}\expo{N}}

&=

\sum_{\ell = 0}^{\infty}{M^{\ell} \over \ell!}

\sum_{\ell’ = 0}^{\infty}{N^{\ell’} \over \ell’!}

=

\sum_{\ell = 0}^{\infty}\sum_{\ell’ = 0}^{\infty}{M^{\ell}N^{\ell’} \over \ell!\ell’!}

\sum_{n = 0}^{\infty}\delta_{n, \ell + \ell’}

\\[3mm]&=

\sum_{n = 0}^{\infty}\sum_{\ell = 0}^{\infty}{M^{\ell} \over \ell!}

\sum_{\ell’ = 0}^{\infty}{N^{\ell’} \over \ell’!}\,\delta_{\ell’,n – \ell}

=

\sum_{n = 0}^{\infty}

\sum_{\ell = 0 \atop {\vphantom{\LARGE A}n – \ell\ \geq\ 0}}^{\infty}

{M^{\ell} \over \ell!}\,{N^{n – \ell} \over \pars{n – \ell}!}

\\[3mm]&=

\sum_{n = 0}^{\infty}{1 \over n!}

\sum_{\ell = 0}^{n}

{n! \over \ell!\pars{n – \ell}!}\,M^{\ell}N^{n – \ell}

=

\sum_{n = 0}^{\infty}{1 \over n!}

\sum_{\ell = 0}^{n}{n \choose \ell}M^{\ell}N^{n – \ell}

\\[3mm]&

=\sum_{n = 0}^{\infty}{1 \over n!}\pars{M + N}^{n}

=

\color{#0000ff}{\large\expo{M + N}}\,,\qquad\mbox{since}\quad\bracks{M,N} = 0

\end{align}

Hint.

$$(M+N)^2=M^2+MN+NM+N^2=(\text{if}~~[M,N]=0)=M^2+2MN+N^2.$$

Use this fact in the expansion of $\exp(M+N)$ to arrive at sum of monomials of the form $M^qN^r$ with $q,r\geq 0$ and rational coefficients (without $[M,N]=0$ you would have also monomials of the form $N^rM^q$!) . To finish the proof you need to collect the monomials of the same degree in $\exp(M)\exp(N)$, where the ordering is clear by definition.

The product of two series (one of which is absolutely convergent) is

$$\left(\sum_{n=0}^\infty a_n\right)\left(\sum_{n=0}^\infty b_n\right)=\sum_{n=0}^\infty

\left(\sum_{k=0}^n a_{n-k}b_{k}\right).$$

Applying this to the series,

$$e^Me^N = \left(I + M + \frac{M^2}{2!} + \frac{M^3}{3!}\ldots\right) \left(I + N + \frac{N^2}{2!} + \frac{N^3}{3!} \ldots\right)\\=I+ (MI+IN)+\left(\frac{M^2}{2}I+MN+I\frac{N^2}{2}\right)+\ldots$$

Now compare this to the other sum

$$e^{(M+N)} = I+ (M+N) + \frac{(M+N)^2}{2!} + \frac{(M+N)^3}{3!} + \ldots$$

- Prove that $\zeta(4)=\pi^4/90$
- Arrangement of Numbers
- What broad families are there with non-constant $z$ such that $_2F_1(a,b;c;z)$ is an algebraic number?
- Fundamental Theorem of Calculus.
- Do the two limits coincide?
- How to find a generator of a cyclic group?
- Can I ever go wrong if I keep thinking of derivatives as ratios?
- Why do we study prime ideals?
- Do we know a number $n\gt 5$ with no twin prime $n\lt q\lt 2n$?
- T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
- Is some thing wrong with the epsilon-delta definition of limit??
- Diophantine applications of Spec?
- Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$.
- Inequality between two sequences preserved in the limit?
- Radius of Convergence of $\sum ( \sin n) x^n$.