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

[A.Magnus]

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

 

[I like Serena]

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)

 

[A.Magnus]

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