All Questions
Tagged with quadratic-forms linear-algebra
84 questions
22
votes
1
answer
13k
views
Non-diagonalizable complex symmetric matrix
This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
- 23.3k
20
votes
3
answers
1k
views
Simultaneous "orthonormalization" in $\mathbb{C}^4$
Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix.
So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
- 42.8k
15
votes
2
answers
1k
views
Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
- 45k
12
votes
1
answer
902
views
Positive 4-form
Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 45k
12
votes
2
answers
3k
views
On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 895
11
votes
0
answers
158
views
Characterization of certain 4-dimensional lattices
Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$.
The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
- 11.9k
10
votes
2
answers
2k
views
When is a bilinear form equivalent to a trace form?
Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...
- 1,142
10
votes
1
answer
2k
views
Over which fields is the Sylvester law of inertia valid?
Short version:
Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?
Long version:
Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of ...
- 23.3k
8
votes
3
answers
431
views
Characterising semi-definite positiveness on vectors with non-negative entries
My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this ...
- 1,352
8
votes
1
answer
2k
views
A spectral inequality for positive-definite matrices
Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \...
- 13.5k
8
votes
0
answers
1k
views
roots of quadratic forms
This may be a very silly question, but I was wondering what is known about the roots of a quadratic form over variables $x_1,\ldots,x_n,y_1,\ldots,y_m$ in the finite field $\mathbb{F}_p$. I'm not ...
- 131
7
votes
3
answers
2k
views
Optimization problem on trace of rotated positive definite matrices
Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...
- 362
6
votes
1
answer
623
views
A curious determinant of quadratic forms
In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$.
$$S(a,X)=\...
- 52.3k
5
votes
1
answer
2k
views
The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices? [closed]
Problem:
Suppose we have two real, symmetric and positive definite square matrices $A$ and $B$, i.e.,
$$A_{ij}, B_{ij}\in \mathbb{R}$$
$$A^T=A$$
$$B^T=B$$
$$x^TAx>0 \forall x$$
$$x^TBx>0 \...
- 89
5
votes
3
answers
3k
views
A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure
Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & \...
5
votes
1
answer
342
views
Is the map $GL_n(\mathbb{Z})\to GL_n(\mathbb{Z}/2\mathbb{Z})$ surjective?
Suppose $F$ is a field. I want to know whether the map $GL_n(GW(F))\to GL_n(W(F))$ is surjective, where $GW$ means Grothendieck-Witt and $W$ means Witt. In the case $F$ is algebraic closed, it reduces ...
- 918
5
votes
1
answer
275
views
Are stably equivalent quadratic forms over Z equivalent?
Let $Q_1, Q_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q_1 \oplus R \cong Q_2 \oplus R$ as quadratic forms. Is it necessary that $Q_1 \cong Q_2$?
I know that by Witt's theorem it is true ...
- 2,178
5
votes
1
answer
745
views
Decide how many non-negative solutions a set of multivariate quadratic equations have
Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?
Some explanations:
All the coefficients are real numbers.
The number ...
- 183
5
votes
1
answer
183
views
Identification of conformal classes of pos def quadratic forms on R^2 with unit ball
One of the lemmas at the foundation of Teichmuller theory is as follows. Let $Q(x,y)$ be a positive definite quadratic form. Then there exists unique $\lambda \in \mathbb{R}$ and $\mu \in \mathbb{C}$...
- 53
5
votes
0
answers
237
views
Linearly independent quadratic forms vanishing on a finite set of points
The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots,...
- 26.9k
5
votes
0
answers
596
views
Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
- 1,421
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 43
4
votes
1
answer
544
views
Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
- 1,371
4
votes
1
answer
393
views
Can an ellipsoid be moved freely inside another ellipsoid?
An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
- 87
4
votes
3
answers
1k
views
Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
- 143
4
votes
1
answer
277
views
Is there a good notion of kernels of quadratic forms on abelian groups?
Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
- 1,813
4
votes
0
answers
216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
4
votes
0
answers
390
views
Fully Homomorphic Error Correction?
Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x=...
- 3,979
4
votes
0
answers
835
views
Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
- 41
3
votes
2
answers
1k
views
Roots of quadratic vector equation
Given $$A_{i j k}X_j X_k + B_{ij} X_j + C_i = 0$$ where $A_{ijk}$, $B_{ij}$, and $C_i$ are arbitrary real numbers for all $i$, $j$, $k$ which are $N$-dimensional indices, such that $A_{ijk}=A_{ikj}$ ...
- 151
3
votes
1
answer
250
views
action of SO(q)
Let $(V,q)$ be a non-degenerate quadratic space. Then we know that for any $d$ with $0 \leq 2d \leq \dim V$, the group $O(q)$ of isometries of $(V,q)$ acts transitively on the set of totally isotropic ...
- 5,163
3
votes
3
answers
590
views
Solving diagonal simultaneous quadratic equations
A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
- 131
3
votes
1
answer
600
views
« Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices
I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, ...
- 1,026
3
votes
0
answers
138
views
Is there a method to solve a non-linear quadratic matrix equation?
I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$
Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
- 31
3
votes
0
answers
107
views
pavings and quadratic forms
Hi,
let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$.
An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...
- 31
2
votes
3
answers
348
views
if Y-X is positive semi-definite, are the eigenvalues of Y bigger?
So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\...
- 87
2
votes
1
answer
747
views
Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?
Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$.
In "Iskovskikh ...
- 1,833
2
votes
2
answers
1k
views
Rank of a linear combination of quadratic forms
Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
- 375
2
votes
1
answer
2k
views
Maximum dimension of an isotropic subspace in a quadratic space
i hope my question is not too trivial.
Let's suppose we have a vector space $V$ with a unimodular quadratic form $q$ of signature $(m,n)$.
My question is: which is the maximum dimension of an ...
2
votes
2
answers
262
views
on the determination of a quadratic form from its isotropy group in char. 2
So this question is a continuation of the following one
[1] On the determination of a quadratic form from its isotropy group
For some motivations and relevant backgrounds related to this question ...
- 7,609
2
votes
2
answers
323
views
Determinants of minors occurring 'within' determinant of full matrix
$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...
- 675
2
votes
2
answers
468
views
Orthogonal transformations fixing a subspace (setwise)
Let $(V,Q)$ be a non-degenerate quadratic space of dimension $n$ over an algebraically closed field of characteristic zero. Let $W$ be a subspace of $V$ of dimension $m < \frac12 n$ which is ...
- 5,163
2
votes
2
answers
119
views
Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
- 947
2
votes
1
answer
210
views
Gram matrix modulo 4
Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and ...
- 623
2
votes
0
answers
55
views
Quadratic surjective map between spheres
The quadratic function $f:\mathbb R^4\to\mathbb R^3$
$$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$
surjectively maps the sphere $S^3$ to the sphere $S^...
- 479
2
votes
0
answers
266
views
Solutions to the quadratic matrix equation $X A X^T = B$
Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric, positive-semidefinite, full-rank matrices. I would like to understand the set of $X \in \mathbb{R}^{n \times n}$ which are themselves symmetric and ...
- 453
2
votes
0
answers
114
views
Projective group of Plucker quadric over the reals
A somewhat elementary question but seemingly difficult to find a suitable reference:
Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
- 356
2
votes
0
answers
111
views
Inseparable field extensions of degree p and linear independence
Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...
- 21
2
votes
0
answers
47
views
Generalization of positive definiteness for a non-smooth quadratic form
I have a "quadratic form" defined as follows:
$$f(\vec x) = \sum_{i<j} c_{ij} x_i x_j + \sum_i a_i(x_i) x_i^2$$
where the coefficients $c_{ij}$ are fixed constants, but the coefficients $a_i(x_i)$...
- 884
2
votes
0
answers
108
views
Angle between subspaces in indefinite signature
I know there is a notion of principal angles between subpaces of a Euclidean space, but what about subspaces of a finite dimensional vector space $V$ equipped with a non-degenerate quadratic form of ...
- 452