1
vote
3answers
55 views

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is a general ...
0
votes
0answers
19 views

Changing a negative definite matrix to a positive definite matrix

Consider a negative definite matrix $X$,then $(I-e^X)$ is a positive definite matrix. What condition should matrix $X$ satisfy?
0
votes
1answer
28 views

Making a matrix invertible

Given $N$ distinct real numbers $x_1,\ldots, x_N$, how can I show that there exist real numbers $a_1,\ldots, a_N$ so that the following matrix is invertible? $$\begin{bmatrix} \exp(ia_1 x_1) & ...
3
votes
1answer
104 views

When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate?

Let $C$ be a $2 \times 2$ asymmetric matrix with real entries. Assume that $C$ has strictly negative, real eigenvalues. Fix $D\in\mathbb{R}^2$, where $D > 0$ (i.e., both coordinates are strictly ...
0
votes
2answers
54 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
6
votes
1answer
83 views

Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.

We know that if two matrix $A$ and $B$ commutes then $\exp(A+B)=\exp(A)\exp(B)$. I am trying to find two matrix that does not commute but $\exp(A+B)=\exp(A)\exp(B)$ is true for them. Can anybody give ...
0
votes
1answer
21 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
3
votes
1answer
37 views

Skew-symmetric matrix and exp function $e^A$

Let $A_{nXn}(\mathbb{R})$ Skew-symmetric matrix $A=-A^t$ prove that $e^A(e^A)^t=I$ while: $e^A=\sum_{i=0}^{\infty} \frac{A^n}{n!}$ I tried this: $A=-A^t \Rightarrow A$ is Diagonalizable with ...
4
votes
2answers
89 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
2
votes
1answer
90 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
1
vote
0answers
83 views

How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
4
votes
2answers
96 views

Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
0
votes
0answers
70 views

A question about exponential matrices

So here is my question, I would like to prove, If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$ And here is ...
3
votes
2answers
82 views

Quick Fact Check if $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$?

If $A$ and $B$ Commute, $\exp((A+B)t)= \exp(At)\cdot\exp(Bt)$? is this statement true?
10
votes
1answer
187 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
9
votes
2answers
917 views

Integral of matrix exponential

Let us be given a square $n \times n$ matrix $A$. For a system \begin{align*} \dot{x}(t) = A x(t), \hspace{0.3 cm} x(0) = x_0 \end{align*} the solution is given by $x(t) = e^{At} x_0$. I am ...
1
vote
1answer
43 views

Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $

The question is Find $\exp(D)$ where $D = \begin{bmatrix}5& -6 \\ 3 & -4\end{bmatrix}. $ I am wondering does finding the $\exp(D)$ requires looking for the canonical form... Could ...
2
votes
3answers
139 views

Find $\exp(D)$ where $D = \begin{bmatrix}2& -1 \\ 1 & 2\end{bmatrix}. $

$$C = \begin{bmatrix}2& -1 \\ 0 & 2\end{bmatrix}\quad $$ I break it down into two matrices $$A = \begin{bmatrix}2& 0 \\ 0 & 2\end{bmatrix}\quad \text{and}\quad B =\begin{bmatrix}0 ...
1
vote
1answer
187 views

Find square matrices $A, B$, such that $\exp(A + B) \neq \exp(A) \exp(B)$.

The question is as the Title stated: I picked a very easy example. However, I am afraid, I am missing something. The two matrices that I picked are $$A = \begin{bmatrix}1& 0 \\ 0 & ...
1
vote
2answers
141 views

Why is $\det(e^X)=e^{\operatorname{tr}(X)}$? [duplicate]

I've seen on Wikipedia that for a complex matrix $X$, $\det(e^X)=e^{\operatorname{tr}(X)}$. It is clearly true for a diagonal matrix. What about other matrices ? The series-based definition of exp ...
11
votes
6answers
355 views

Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
6
votes
3answers
216 views

Prove that $e^{-A} = (e^{A})^{-1}$

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I've tried messing around with both sides, evaluated as sums. I just can't get the two to match up. Any ...
2
votes
2answers
292 views

Not commuting exponential matrices

Reading this book I came across the following formula : $$ e^A e^B = e^{A+B}e^{\frac{1}{2}[A,B]} $$ where $A$ and $B$ are two matrices and $[A,B] = AB-BA$. I tried to find a demonstration without ...
1
vote
1answer
270 views

Bound on the norm of a matrix exponential in Jordan Form

I'm looking to prove the following lemma: Let $A$ be a matrix in $\mathbb{R}^{n\times n}$. Then for any $\lambda^* > \max_{\lambda} \; \mathrm{Re} \; (\lambda)$ such that $ \lambda \in\sigma (A)$, ...
3
votes
1answer
89 views

Easy proof that $\exp{Xt} = I \Rightarrow X = 0$

Let $X\in \mathbb{C}^{n\times n}$ and $I$ is identity matrix , than if: $$ \forall t\in \mathbb{R}\quad e^{Xt} = I $$ than $$ X = 0. $$ I'm looking for short and slick proof of this ...
0
votes
1answer
77 views

Matrix integral of absolute exponential item

If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral? $$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$ where ...
0
votes
1answer
135 views

Matrix Exponential equality

I was reading about the matrix exponential function and I came across this: If $xy = yx$ then $$ \exp(x+y) = \exp(x)\cdot\exp(y) $$ My textbook gives a proof as follows: $$ \exp(x+y) = ...
0
votes
1answer
95 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
2
votes
0answers
82 views

Inverse function of product of exponential matrices

I am looking for the value of $\mathbf{X}$ in a function of the type \begin{align} (\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B} \end{align} where ...
3
votes
3answers
55 views

$ e^{At}$ for $A = B^{-1} \lvert \cdots \rvert B $

For a homework problem, I have to compute $ e^{At}$ for $$ A = B^{-1} \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} B$$ I know how to compute the result ...
2
votes
1answer
451 views

Commuting in Matrix Exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$. Let \begin{equation} \exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i \end{equation} show that $\exp(A+B) = \exp(A).\exp(B)$.
0
votes
1answer
85 views

Definition of $\exp(A)$ in terms of spectral decomposition.

I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$ What ...
10
votes
2answers
891 views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
4
votes
2answers
404 views

Diagonalise a matrix and show the formula

I have diagonlised P to get $$P=\left(\begin{matrix} -1 &0 &0\\ 0 &0 &0\\ 0 &0 &1 \end{matrix}\right)$$ however am unsure on how to proceed, would appreciate any help! By ...
3
votes
0answers
97 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
3
votes
0answers
166 views

Infinite series of matrices almost but not quite matrix exponential

I'm working on a problem that has brought up for me the need to address infinite series of the following form, $$ \sum_{i=k}^\infty \frac{1}{i!}A^{i-k+1} $$ where $A$ is an $n\times n$ matrix. If $k = ...
2
votes
1answer
513 views

Fundamental matrix and exponential of matrix using Laplace Transform

I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$ I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$ at ...