# Tagged Questions

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### Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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)
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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!
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