MHB Index of Positivity in Quadratic Form: $f(X) = \sum^{n}_{i=1} x_{2}^{i}$

  • Thread starter Thread starter A.Magnus
  • Start date Start date
  • Tags Tags
    Index
A.Magnus
Messages
138
Reaction score
0
I am working on this problem: Show that the index of positivity is $n$ for this quadratic form:

$$f: \mathbb R^n \rightarrow \mathbb R, \ f(X) = \sum^{n}_{i=1} x_{2}^{i} + \frac{1}{n} \sum_{i \neq j}x_ix_j.$$

Here is a solution I got from other sources: Since the index of quadratic form is the number of positive terms (or square terms) of its canonical form, and since the canonical form of $f$ is

$$\big(x_1^2 + x_2^2 + ... + x_n^2 \big)+ \frac{1}{n}\big((x_1x_2 + x_1x_3 + ... + x_1x_n) + (x_2x_3 + x_2x_4 + ... + x_2x_n) + ... \big),$$

therefore the index is $n$.

I have strong doubt that this solution is correct. Is this correct? How do I have to go about if this is wrong? Thank you for your gracious help and time. ~MA
 
Last edited:
Physics news on Phys.org
MaryAnn said:
I am working on this problem: Show that the index of positivity is $n$ for this quadratic form:

$$f: \mathbb R^n \rightarrow \mathbb R, \ f(X) = \sum^{n}_{i=1} x_{2}^{i} + \frac{1}{n} \sum_{i \neq j}x_ix_j.$$

Here is a solution I got from other sources: Since the index of quadratic form is the number of positive terms (or square terms) of its canonical form, and since the canonical form of $f$ is

$$\big(x_1^2 + x_2^2 + ... + x_n^2 \big)+ \frac{1}{n}\big((x_1x_2 + x_1x_3 + ... + x_1x_n) + (x_2x_3 + x_2x_4 + ... + x_2x_n) + ... \big),$$

therefore the index is $n$.

I have strong doubt that this solution is correct. Is this correct? How do I have to go about if this is wrong? Thank you for your gracious help and time. ~MA

Hey MaryAnn! ;)

I didn't answer yet because I'm not familiar with this index of positivity, and I couldn't find any references to it.
Either way, with the definition you gave, it seems to me that you applied it correctly. (Nod)
 
I like Serena said:
Hey MaryAnn! ;)

I didn't answer yet because I'm not familiar with this index of positivity, and I couldn't find any references to it.
Either way, with the definition you gave, it seems to me that you applied it correctly. (Nod)

Thank you. Let me know if you find any reference to this one. Thanks again for your gracious help. ~MA
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top