Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [kronecker-symbol]

For questions on kronecker symbols, a generalization of the Jacobi symbol to all integers.

In number theory, the Kronecker symbol, written as ${\displaystyle \left({\frac {a}{n}}\right)}$ or ${\displaystyle (a|n)}$, is a generalization of the Jacobi symbol to all integers.

57 questions
8
votes
0 answers

Application of Legendre, Jacobi and Kronecker Symbols

Legendre, Jacobi and Kronecker Symbols are powerful multiplicative functions in computational number theory. They are useful mathematical tools, essentially for primality testing and integer factorization; these in turn are important in…
Ali Abbasinasab
  • 1,783
6
votes
1 answer

Confusion about the Kronecker $\delta$

Something disturbs me, concerning the Kronecker $\delta$. Assuming these hold: $$\delta_{ij}\delta_{jk}=\delta_{ik}$$ $$\delta_{ij}=\delta_{ji}$$ $$\delta_{ii}=1$$ does it follow that for every $\delta_{ij}$ we have…
Whyka
  • 2,003
4
votes
2 answers

Proof of $\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

I'm a student of physics. There is an identity in tensor calculus involving Kronecker deltas ans Levi-Civita pseudo tensors is given by $$\epsilon_{ijk}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ which is extensively used in…
SRS
  • 1,532
4
votes
1 answer

Prove that $x\frac{d}{dx} \delta (x) = -\delta (x)$

Is this proof formal enough? I plan on being a theoretical physicist one day, so I want to get into the good habit of being mathematically strict. My proof: $u=x$; $du=dx$ $v = \delta (x)$; $dv = -\delta (x)$ $$\int x \frac{d}{dx}(\delta (x))dx =…
whatwhatwhat
  • 1,657
4
votes
2 answers

Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven't quite pinned down why it works or what its…
Matt Miller
  • 51
4
votes
2 answers

Why is the delta function the continuous generalization of the kronecker delta and not the identity function?

In a discrete $n$ dimensional vector space the Kronecker delta $\delta_{ij}$ is basically the $n \times n$ identity matrix. When generalizing from a discrete $n$ dimensional vector space to an infinite dimensional space of functions $f$ it seems…
asmaier
  • 2,763
3
votes
2 answers

Levi Civita and Kronecker Delta

I've been working on some quantum mechanics problems and arrived to this one where I have to deal with subscripts. I got stuck doing this: I have $\epsilon_{imk}\epsilon_{ikn}=\delta_{mk}\delta_{kn}-\delta_{mn}\delta_{kk}$. But then I went to check…
RicardoP
  • 221
3
votes
0 answers

Proof that Kronecker Delta is Mixed Tensor

The book I am reading asks the reader to verify that the Kronecker Delta is a second-order mixed tensor with one contravariant and one covariant index as indicated: $$ \delta_j^i = \left\{\begin{array}{ll} 1 & i = j\\ 0 & i \neq…
Tim Clark
  • 418
  • 1
  • 3
  • 7
3
votes
1 answer

The Relation Between Kronecker's Delta and the Permutation Symbol

The Kronecker's Delta is defined as $$\delta_{ij}= \begin{cases} 1 & i=j \\ 0 & i \ne j \end{cases}$$ Also, the Permuation Symbol known as Levi Cevita's Symbol is introduced as $$\varepsilon_{ijk}= \begin{cases} 1 & \text{$ijk$ is an even…
Hosein Rahnama
  • 15,554
3
votes
0 answers

Clarify the definition of a transition matrix

I am reading the book Matrix Variate Distribution by A. K. Gupta and D. K. Nagar. In the first chapter (Definition 1.2.8), they define a matrix $B_{p}$ ($p \in \mathbb{N}^{\ast}$) as follows : Definition 1.2.8 : The matrix $B_{p}$ of order $p^{2}…
Odile
  • 1,171
  • 8
  • 11
3
votes
1 answer

Why does the Kronecker symbol $\left(\frac{a}{\cdot}\right)$ define a character?

Let $a\not\equiv 3\pmod 4$ and $a\ne 0$. How do you show that $\chi(n):=\left(\frac{a}{n}\right)$ (where $\left(\frac{\cdot}{\cdot}\right)$ denotes the Kronecker symbol) defines a character of modulus $m:=4|a|$ if $a\equiv 2 \pmod 4$ and $m:=|a|$…
principal-ideal-domain
  • 6,324
2
votes
1 answer

Product of two Levi-Civita permutation symbols

I am attempting to follow the solution provided by (Mark Viola) here (Kronecker delta and Levi-Civita symbol). I do not understand the following: Step 1 & 2 where the initial Levi-Civita symbols are expanded from only 1 term into 3. I am assuming…
user825535
  • 25
2
votes
2 answers

name or symbol for "anti" Kronecker delta?

Is there a name or symbol convention for what I might call the "anti" Kronecker delta (that is, $1 - \delta_{ij}$)?
Grayscale
  • 205
2
votes
1 answer

Kronecker Delta Expressions

I am trying to understand the Kronecker Delta and want to clarify. Considering the definition of the Kronecker Delta and assuming $i=j=k$ for the following situations: I know that $\delta _j^i \delta _i^j $ is equal to $N$ where $N$ is the dimension…
Cave Johnson
  • 35
2
votes
0 answers

Simplify bra-ket notation with kronecker product and kronecker sum

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation. I have the following equation(Swap gate with 3 cnot): First equation $|\phi_2\rangle = |i\bigoplus(i\bigoplus…
Levente
  • 143
1
2 3 4