# Tagged Questions

Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

1answer
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### Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
2answers
396 views

### Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
2answers
26 views

### What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology?

I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a ...
1answer
33 views

### Confusion of a formula about Lagrangian

Recently, I am reading a paper about eigenvalue problems. Consider the following problem, which occurs at the first page of the paper. \begin{align} \text{minimize}\quad &x^TAx \\ \text{subject ...
2answers
1k views

### Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \...
1answer
2k views

0answers
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### Linear transformation of positive definite diagonal matrix

Let $\mathbf \Psi$ denote the set of all positive definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Phi$ denote the set of all positive semi-definite, diagonal, nXn dimensional,...
0answers
30 views

### Is sum (convex combination) of quadratic function/aggregator quadratic?

We know convex combination of concave/convex functions are concave/convex. While convex combination of two quasi-convex/quasi-concave functions necessarily quasi-convex/quasi-concave. Common ...
1answer
31 views

2answers
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### Describe an equation geometrically

Finishing the last few stuff left for my end-term semester exams on Linear Algebra II, I bumped across a collection of identical exercises, posting one below : Describe geometrically, giving as much ...
0answers
48 views

1answer
31 views

2answers
28 views

### How do I convert a quadratic form to a diagonal form?

I don't understand how I should choose the transformations to convert a quadratic form to a diagonal form. Ex: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
1answer
90 views

### How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$f(x) = \frac{1}{2}x^T A x - b^Tx + c$$ My question is: How can we deduce this formula? I understand, ...
0answers
22 views

### Quaternary quadratic modular problem.

Consider quadratic form $$Q(w,x,y,z)=w^2-x^2-y^2+z^2$$ and fix $r\in(0,\frac12)$ and pick a large enough $n\in\Bbb N$. How do we find a solution to $$Q(w,x,y,z)\bmod n=0$$ on condition that \sqrt n\...