1
vote
1answer
37 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
-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
0answers
10 views

An extension of the Golden-Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality is correct, at least in some cases: $$tr\left(A e^{B+C} \right) \leq tr\left(A ...
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
3answers
57 views

Finding the matrix exponential

Find the matrix exponential of $$\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}.$$ Since this matrix is not diagonalizable, you will have to use the definition of the matrix exponential. ...
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 ...
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? ...
1
vote
1answer
71 views

Matrix to Matrix Power [duplicate]

Considering $A^b$, we have a nice definition and properties when $A$ is a matrix and $b$ is a real (or field) value. We also have a fairly useful definition for $e^B$, where $B$ is a matrix and $e$ ...
5
votes
2answers
291 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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 ...
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
2answers
91 views

Checking inequality without actually calculating LHS and RHS

How to check whether the following inequality is true or not without actually calculating the values of $x^y $ and $y^x $: $$ x^y > y^x$$ (x and y are integers)
2
votes
1answer
59 views

How to find $x + y + z$?

Q. If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then (A) $x + y + z = 3 xyz$ (B) $x + y + z = 0 $ (C) $( x + y + z)^3= 27 xyz$ (D)$ x^3 + y^3 + z^3 = 0$ What I've done: ...
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 ...
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
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
454 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
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: ...
0
votes
0answers
200 views

Approximate a complicated mystery function

Let there exist a mystery function ƒ. ƒ accepts exactly 2 arguments, A & B. As B approaches A, ƒ approaches A, at a simple exponential growth rate E. As B approaches 0, ƒ approaches the mean ...
0
votes
1answer
43 views

Difficulty understanding some algebra done in a problem

The main problem is about computer science, trying to show that $f(x)=e^{x^Tx'}$ is of the form $\exp{\Big( \frac{||x - x'||^2}{2\sigma^2} \Big) }$, so it could be a kernel function. (see here for ...
0
votes
1answer
18 views

How to find an algebraic term when 2 similar exponential forms of it are given?

if $a^2bc^3 = 25$ and $ab^2= 5$, how would you find $abc$? Is there a formula of which these are a part of? What I've done is this: $$abc^3 = \frac{25}a\quad\implies\quad abc = ...
1
vote
2answers
592 views

Solving system of recurrence relations

Base Case: $$ \left\{ \begin{array}{c} T(1) = 1 \\ T(2) = 1 \\T(3) = 4\end{array} \right. $$ I have the system: $$ \left\{ \begin{array}{c} T(N) = G(N-1) + F(N-1) \\ G(N) = F(N-1) + G(N-1) \\ ...
6
votes
2answers
639 views

A “Matrix Trigonometry”

$e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix): $$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$ A thought occurred to me that ...
9
votes
1answer
901 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} & ...
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 ...
0
votes
1answer
1k views

Can a variable appear as a denominator in a linear equation or is the system non-linear?

I am trying to figure out whether the following equation is non-linear or if it's linear, how would I solve it? $x+\frac{2}{y}=0$ It can be rewritten as $x+y^{-2}=0$ so I guess if this is ...
5
votes
1answer
240 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...