1
vote
0answers
14 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
-1
votes
1answer
28 views

Special cases of the $A^k$ matrix power

Let $A$ be az $n \times n$ real matrix. Suppose, that there is a fixed $k_0 \in \mathbb{N}_+$, and for this $k_0$ the matrix power $A^{k_0}$ has a closed formula, for example $A^{k_0}=k_0 A$ or ...
0
votes
1answer
20 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 ...
0
votes
1answer
19 views

Exponent of polynomials (of matrices)

$A$ is a matrix over $\mathbb R$ (reals). Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$ I tried using the sigma writing but got stuck (I ...
1
vote
1answer
39 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
0
votes
0answers
36 views

Quick question on matrix exponentiation.

Hi can someone explain a step in the proof that if A, B are commutative matrices then $e^Ae^B=e^{A+B}$ So define $f(t)=e^{tA}e^{tB}$ then $f(0)=I$ and we have that $f'(t)=Ae^{tA}e^{tB}+e^{tA}Be^{tB}$ ...
5
votes
2answers
185 views

Prove: matrix A is diagonalizable iff exp(A) is diagonalizble

I need to prove: matrix A is diagonalizable iff $\exp(A)$ is diagonalizble. exp means exponent function. I know to prove that if $A$ is diagonalizable so $\exp(A)$ is diagonalizable, but have a ...
2
votes
0answers
32 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
1
vote
1answer
62 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
1
vote
2answers
65 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
3
votes
1answer
48 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now ...
1
vote
2answers
80 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
10
votes
1answer
181 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 $n\times n$ ...
3
votes
2answers
71 views

Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective?

This is related to a personal exploration of isometries of directed graphs, motivated by my son's Lego Duplo train tracks and identifying "interesting" layouts. If $M$ is the adjacency matrix for a ...
2
votes
3answers
45 views

Why can matrix exponentiation be done by squaring?

Matrix multiplication is not communative: A*B != B*A Then why can matrix exponentiation be done by squaring? I have tried searching for special cases where this rule did not apply, but from what I've ...
0
votes
2answers
418 views

Evaluate a matrix with a negative power

I am having problem with how to calculate a matrices that are raised to negative powers. I can manage the adding, multiplication etc, but I am stuck here. The matrix in question is ...
0
votes
0answers
27 views

$A_{2}^{T} + A_{2} < 0$ for $A_{2} =(A_{1}A_{0}^{-1})^{\alpha}A_{0}$?

Given are two matrices $A_0, A_1$, whose symmetric part is negative definite: $A_{0}^{T} + A_{0} < 0$, $A_{1}^{T} + A_{1} < 0$ Proof that: $A_{2}^{T} + A_{2} < 0$ for $A_{2} = ...
1
vote
3answers
72 views

How to prove that $\|e^{X+Y}-e^X\| \leq \|Y\| e^{\|X\|} e^{\|Y\|}$?

A couple of questions from the Wikipedia "matrix exponential" article: In the part of the article I linked to, they mention that to conclude that every matrix in $GL(n)$ has a logarithm (though not ...
1
vote
1answer
46 views

Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are: It mentions that Lieb's thoerem "accomplishes ...
5
votes
4answers
78 views

Convergence of exponential matrix sum

Let $A$ be an $n\times n$ matrix. Consider the infinite sum $$B=\sum_{k=1}^\infty\frac{A^kt^k}{k!}$$ Each term $\dfrac{A^kt^k}{k!}$ is an $n\times n$ matrix. Does the sum $B$ always converge? (i.e. ...
2
votes
1answer
57 views

What is the exponential for the matrix

What is the exponential for the matrix $$ \begin{pmatrix} 0 & -x & 0 \\ x & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$ Is it $$ ...
1
vote
0answers
46 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
8
votes
1answer
158 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
1
vote
1answer
41 views

About a matrix, its powers, and a particular value…

Is it possible to find two matrices, $A$ an $B$ such that: $$A B^0 x = \begin{pmatrix} c_0 \\ ? \\ ? \\ \end{pmatrix}, $$ $$A B^1 x = \begin{pmatrix} c_1 \\ ...
1
vote
1answer
65 views

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this true when $A$ is a matrix?

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this property true when $A$ is a matrix? Suppose $$A=\begin{pmatrix} 1 &0 &1\\ 1 &0 &0\\ 0 &1 &0 ...
0
votes
1answer
91 views

Is the operation taking a matrix to the power of another matrix well-defined?

e.g. if A and B are matrices, is there a useful definition for $A^B$? I don't see an obvious definition; but then the definition of the matrix exponential also would never occur to me independently, ...
2
votes
8answers
412 views

Do matrices have a “to the power of” operator?

Well I was sure that saying "$A^3$" (where $A$ is an $n\times n$ matrix) is nonsense. Sure one could do $(A\cdot A) A$ But that contains different operators etc. So what did my prof mean by the ...
0
votes
1answer
138 views

Is there fast approximation of the n-th power of diagonalizable matrix A?

My thoughts on the subject. Because of diagonalizability $A$ can be written as $A = PDP^{-1}$ and then $A^n = PD^nP^{-1}$ here $P$ is matrix of eigenvectors and $D$ is diagonal matrix with ...
11
votes
6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
0
votes
1answer
91 views

k-fold matrix product

For $k \in \mathbb{N}$, $B,C \in \mathbb{R^{n,n}}$, given the matrices $B,C$ , calculate all powers $B^k$ and $C^k$ I'm a bit puzzled by this task. I assume it's supposed to practice handling ...
2
votes
1answer
101 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
1
vote
0answers
58 views

How to solve for the matrix in a set of equations involving the matrix exponential?

I was wondering how to solve the following problem (in a least-squares sense): $$ \mathbf{y}_1 = e^{Ax_1} \mathbf{y}_0 \\ \mathbf{y}_2 = e^{Ax_2} \mathbf{y}_0 \\ \vdots\\ \mathbf{y}_n = e^{Ax_n} ...
4
votes
4answers
578 views

Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
1
vote
2answers
526 views

Trace of the matrix power

Say I have matrix $A = \begin{bmatrix} a & 0 & -c\\ 0 & b & 0\\ -c & 0 & a \end{bmatrix}$. What is matrix trace tr(A^200) Thanks much!
1
vote
2answers
2k views

Raising a square matrix to a negative half power

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations: $$S = (\textrm{diag } R^{-1})^{-1/2}$$ I understand the diagonal and inverse ...
1
vote
3answers
126 views

conditions under which real-matrix exponential are equivalent

Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$? Thanks!
3
votes
2answers
455 views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & ...
2
votes
1answer
272 views

Exponential of the integral of a matrix - what is wrong with this calculation/statement?

I'm going through a book we're using in our intermediate differential equations course, and this is one of the problems that are contained in it. Note that this question is not tagged as homework, ...
1
vote
2answers
219 views

$\det(\exp X)=e^{\mathrm{Tr}\, X}$ for 2 dimensional matrices

I want to prove that for $X\in M_2(\mathbb{R})$ the formula $\det(\exp X)=e^{\mathrm{Tr}\, X}$ holds, writing $X$ in normal form gives $X=PJP^{-1}$, where $J$ is the Jordan matrix, now $\exp ...
1
vote
2answers
250 views

Computing exponential of a $2\times 2$ matrix using only its trace and determinant.

I want to compute the exponential of an arbitrary $2\times 2$ matrix over $\mathbb{R}$ only using its trace and determinant. I've shown that for a traceless matrix $A$ there is the following formula: ...
1
vote
2answers
123 views

Failure during calculating the matrix exponential, but where?

I have to calculate $e^{At}$ of the matrix $A$. We are learned to first compute $A^k$, by just computing $A$ for a few values of $k$, $k=\{0\ldots 4\}$, and then find a repetition. $A$ is defined as ...
6
votes
1answer
220 views

Power of a block matrix with eigenvalues on the unit circle

In the expression $$\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}^n = \begin{bmatrix}A^n & * \\ 0 & B^n\end{bmatrix},$$ I wonder whether the term denoted by * can be expressed in a simple ...
12
votes
2answers
552 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
9
votes
1answer
902 views

Matrix raised to a matrix

Good evening, I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is $$ M^M $$ valid, or is there at least something ...
2
votes
1answer
146 views

Has a matrix block diagonal structure if and only if its exponential has it as well?

Obviously if $\mathbf{A}=\begin{bmatrix}\mathbf{C} & \mathbf{0} \\ \mathbf{0} & \mathbf{D}\end{bmatrix}$ then $e^{A}=\begin{bmatrix}\mathbf{e^C} & \mathbf{0} \\ \mathbf{0} & ...
2
votes
2answers
108 views

Using exponents when working with matrices

I was working on the following problem when I stumbled upon an oddity. If $X=P^{-1}AP$ and $A^3=I$, prove that $X^3=I$ My first approach was to cube both side which led to the following: ...
0
votes
1answer
138 views

Calculating $f(x)$ using matrix exponentiation

Having polynomial in form $\sum \limits_{i=1}^{n} c_i x^i$ and linear recurrence, for example $$ R(n)=2 R(n-1)+3R(n-2) + f(n)$$ where $f(n)$ is our polynomial and $R(n)$ is linear reccurence, how ...
7
votes
1answer
2k views

Ways to calculate the derivative of the matrix exponential

Could someone provide me with a rigorous proof as to why the derivative of the function $f:t \ni \mathbb{R} \mapsto e^{tA}\in \textrm{Mat}_n (\mathbb{R})$ is $t \mapsto A\cdot e^{tA}$ ? I didn't ...
1
vote
2answers
741 views

Matrix exponential of a 2x2 matrix composed of a antihermitian matrix and a symmetric matrix

As per the title, I'd like to calculate the exponential of a matrix which has an antihermitian component and a symmetric component (although this fact may not be useful). More specifically ...
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...