0
votes
1answer
26 views

Will a 2 by 2 quadratic form be negative definitive if it has repeated eigenvalues which are negative?

Say we have the quadratic form $$ f = x^T Q x \\ Q = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$$ which has repeated eigenvalues $\lambda = -1$. Will the quadratic form be negative ...
3
votes
1answer
34 views

Does an isotropic vector always exist for an indefinite quadratic forms?

I have some problem reading some paper. In that paper, the author proved that a quadratic form $Q$ has an isotropic vector(of course, nonzero) by showing $Q$ has both nonnegative and nonpositive ...
6
votes
1answer
91 views

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let ...
1
vote
1answer
52 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
0
votes
2answers
29 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
3
votes
2answers
30 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
2
votes
1answer
52 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
1
vote
1answer
49 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
3
votes
1answer
60 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
3
votes
1answer
45 views

Question regarding quadratic form exercise in Hoffman Kunze

In the book the quadratic form associated with a bilinear form f is defined as $q(\alpha)=f(\alpha,\alpha)$. Then, if $U$ is a linear operator on $\mathbb R^2$ an operator $U^\dagger$ on the space of ...
0
votes
1answer
16 views

Expanding the Malhalanobis distance to find sufficient statistics of a multivariate Gaussian distribution.

Given a dat set $X=(x_1,...,x_N)^T$ in which the observations $x_n \in R^D$ are assumed to be drawn independently from a multivariate Gaussian distribution with mean $\mu \in R^D$ and covariance ...
2
votes
1answer
74 views

Quadratic Form - New Axes = Eigenvectors of P, Order of Eigenvectors Important? [Kolman P539 Example 6]

Hypothesise that $P$ is the symmetric matrix of some quadratic form $g(\mathbf{ x} ) = \mathbf{ x^TAx} $. Then $P$ is the orthogonal matrix consisting of orthogonal eigenvectors of $A$. Moreover, use ...
2
votes
3answers
44 views

A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same?

Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form? Positive defined quadratic form is A(x,y), that ...
0
votes
2answers
26 views

How to prove: a quadratic form with a matrix $ B = CC^T $ is positive defined?

Let a matrix $ C \in \Bbb K^{n \mathtt x n} : det(c) \ne 0 $ (K is any field - C or R) $ \Rightarrow $ a quadratic form with a matrix $ B = CC^T $ is positive defined one. How to prove it?
2
votes
1answer
45 views

Law of large numbers for linear (quadratic) combinations of i.i.d. random variables

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as ...
0
votes
1answer
38 views

Concavity of quadratic form

I know that the quadratic form $x'Ax$ is a concave in vector $x$ if matrix $A$ is negative semi definite. What happens if $A$ depends on $x$ (so that I have $x'A(x)x$), but I still know that $A(x)$ is ...
3
votes
1answer
67 views

What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
5
votes
1answer
48 views

Why must P be orthonormal, and not just orthogonal, for change of variable in Quadratic Form? [Kolman P560 8.8.24]

Lay P402 : A change of variable is an equation of the form $x=Py$, where $P$ is an invertible matrix and $y$ is the (neW) coordinate vector of $x$ relative to the basis of $\mathbb{R}^{n}$ determined ...
0
votes
1answer
58 views

On representation of quadratic form

In linear algebra, a quadratic form is defined as $Q(x)=x^TAx$ for some (non-singular) matrix $A$ and any $x\in V$, where $V$ is a vector space. Actually, quadratic form can be any one satisfying ...
2
votes
1answer
36 views

Find two bilinear forms with the same quadratic form over $\mathbb F_2$

Let $V$ be a $K$-vectorspace with a bilinear form $\langle , \rangle$ and the associated quadratic form $q:V \to K, v \mapsto \langle v,v \rangle$. Let $K = \mathbb F_2$. Are there two different ...
2
votes
2answers
68 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
1
vote
2answers
50 views

steepest descent with quadratic form converge in 1 iteration

Well I'm stuck on an exercise given: The steepest descent method is applied to the quadratic form $$Q(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x} + c$$ where $A$, ...
1
vote
1answer
26 views

Perform matrix-vector multiplicacion using quadratic form

I have a quadratic from, $v^{T}Hv +q^Tv+c$, where neither the $H$, $q$ and $c$ are given implicity. The whole expression is very complicated, but it is quadratic form. $H$ is simetric. The question ...
0
votes
0answers
50 views

a question about relationship between KKT matrix equation and optimal solution of quadratic problem.

I have a question regarding how the KKT matrix plays in solving for optimization problem: Is it correct that the optimal solution for quadratic optimization problem with positive definite hessian ...
1
vote
2answers
46 views

Linear Algebra Quadratic True False

"Every quadratic form $x^TAx$ with $A$ an invertible matrix is either positive definite, negative definite, or indefinite." Is this true or false? I am just wondering does it have to be positive, ...
-1
votes
1answer
58 views

Signature of quadratic form and eigenvalues

I'm asking about the signature of the quadratic form - the triple (n0, n+, n−). Is it true that n+ is the number of positive eigenvalues, and n- is the number of negative of eigenvalues of the matrix ...
2
votes
1answer
71 views

Quadratic form and symmetric bi linear form formula, basic point unclear to me

Something really basic but I have to ask it: I was taught that the formula of symmetric bi linear form of the quadratic form if: $f(v,w) = 1/2(q(v+w)-q(v)-q(w))$ but $q$ is linear so what did we get ...
0
votes
4answers
87 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
0
votes
2answers
70 views

Congruent diagonal matrix

For two days I reflect on this question without an answer: If $A=(i+j-1)_{1\le i,j\le n}$ is matrix in $\mathcal M_n(\mathbb R)$, the question is to find basis in which $A$ is congruent to diagonal ...
1
vote
1answer
145 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
0
votes
2answers
108 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
3
votes
1answer
104 views

Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
0
votes
1answer
57 views

Find the signature of the quadratic form

Very simple question but something doesn't make sense to me. We are given a quadratic form (bilinear map but on the same vector twice): $Q(v) = v^t *\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 ...
1
vote
2answers
76 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
0
votes
1answer
53 views

Finding diagonal transformation matrix of a bilinear form

Let $f:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a symmetric bilinear form, and let $q$ be its quadric form, so that $q(x, y, z)= xy+yz$. Find the transformation matrix $A$ of $f$ by ...
0
votes
0answers
38 views

Why is the sign of the determinant of a quadric in $\mathbb R^3$ invariant under affine transformations?

According to my reference, the answer has to do with the fact that the projective closure of a quadric in $\mathbb R^3$ is given by a matrix $\bar A$ of order even (in fact, order 4). But isn't it ...
0
votes
1answer
33 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
1
vote
1answer
44 views

Associated Bilinear Form to Q (Quadratic Form)

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the ...
0
votes
1answer
65 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
0
votes
0answers
113 views

selecting a good upper bound on quadratic form in presence of unknown PD matrix

I have a cost function that is \begin{equation} J=\text{Trace} [\ (\ I-LC)\ KQK^T(\ I-LC)\ ^T ]\ \end{equation} where $Q$ is a unknown positive definite matrix, $K$ and $C$ are full rank $n\times ...
0
votes
1answer
41 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
0
votes
2answers
34 views

Definite Quadratic Form

I got the following problem: Let $V$ be a real vector space and let $q: V \to \mathbb R$ be a real quadratic form, Prove that if the set $L = \{v \in V | q(v) \ge 0\}$ forms a subspace of $V$ then q ...
0
votes
1answer
75 views

Quadratic Forms and their Matrices.

1) How do you manage to transform a matrix from quadratic to canonical form? For instance, assume a linear transformation such that: $$Q(x,y,z)=x^2+2xz+z^2;$$ As far as I can see, in the ...
7
votes
2answers
265 views

How find this matrix $A=(\sqrt{i^2+j^2})$ eigenvalue

let the matrix $$A=(a_{ij})_{n\times n}$$ where $$a_{ij}=\sqrt{i^2+j^2}$$ Question: Find the difference $sign{(A)}$ can see this define:http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia My ...
0
votes
1answer
75 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
0
votes
0answers
94 views

How checking $(-1)^n$ for $H=H_{n+m}$ is equivalent to checking $(-1)^{m+1}$ for $H_{2m+1}$?

This is from "Mathematics from Economists" by Simon and Blume: To determine the definiteness of a quadratic form of $n$ variables, $Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x},$ when restricted to a ...
1
vote
0answers
15 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
0
votes
1answer
64 views

Is $B$ a positive or negative semidefinite?

Let $A$ be an $n\times n$ symmetric matrix. Then, $A$ is a positive semidefinite iff every principal minor of $A$ is $\geq0$; $A$ is a negative semidefinite iff every principal minor of odd order ...
1
vote
3answers
229 views

How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form

We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
0
votes
0answers
198 views

Quadratic form in canonical form

Reduce the quadratic form $q(x,y) = 6xy$ using the orthogonal reduction (i.e, find a orthogonal basis such that the matrix of the bilinear form is diagonal and $a_{ii} = 0$ or $a_{ii} = ^+_-1$) What ...