# Tagged Questions

**3**

votes

**1**answer

182 views

### What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?

If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...

**4**

votes

**0**answers

75 views

### Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...

**5**

votes

**1**answer

279 views

### History of Jordan Canonical Form?

Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:
When and how was it first stated? (I understand it was independently stated ...

**4**

votes

**1**answer

462 views

### A question on Grassmannian

Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...

**7**

votes

**1**answer

469 views

### A question on eigenvalues

Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...

**3**

votes

**1**answer

118 views

### On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...

**3**

votes

**0**answers

131 views

### Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...

**0**

votes

**1**answer

160 views

### when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under ...

**1**

vote

**1**answer

284 views

### Iwasawa Decomposition for Matrices [closed]

I was asked to prove that if
$$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$
denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...

**5**

votes

**1**answer

226 views

### Invariants of a $GL(3,\mathbb{R})$ action

I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...

**3**

votes

**1**answer

136 views

### Reduction of antisymmetric complex matrices

Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...

**4**

votes

**1**answer

208 views

### power log distance between matrices

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...

**8**

votes

**1**answer

541 views

### Unusual decomposition of 3x3 real symmetric matrices - is this possible?

If $M$ is a 3x3, real symmetric matrix, then I know there are a few ways to decompose $M$ as
$M = A^T D A$,
where $D$ is a real diagonal matrix: e.g., this can always be done for some $A \in SO(3)$, ...

**1**

vote

**2**answers

316 views

### Have derivatives of determinants along 1-psg's ever been 'coherently' computed via Jacobi's formula?

Suppose $\mathfrak{p}$ denotes all the symmetric matrices in $\mathfrak{sl}_{2n} \mathbb{R}$.
Then for each parameterized 1-dimensional linear subspace $\xi=\xi(t)$ of $\mathfrak{p}$ we get a ...

**4**

votes

**2**answers

268 views

### Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?

I need to answer (affirmatively, I hope) the following question:
In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset
...

**9**

votes

**1**answer

476 views

### Computing the Zariski closure of a subgroup of SL(n,Z)

Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...

**5**

votes

**1**answer

370 views

### determining symplecticity (if that's a word)

Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the ...

**2**

votes

**1**answer

284 views

### equations over (some) lie groups

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...

**2**

votes

**1**answer

204 views

### An innocent looking subgroup of $U(n)$

Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...

**4**

votes

**3**answers

317 views

### Linear subspaces in cones over orthogonal groups

Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...

**6**

votes

**0**answers

308 views

### Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...

**3**

votes

**4**answers

457 views

### Polar decomposition for quaternionic matrices?

A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...

**0**

votes

**0**answers

109 views

### Generating Set for Og over $\mathbb Z_2$

Hi, All:
I am reading a claim that Og , the orthogonal group associated with a finite-dimensional vector space V over $\mathbb Z_2$ , and a quadratic form q defined therein , i.e., the group of ...

**3**

votes

**3**answers

381 views

### Multiplicity of eigenvalues in 2-dim families of symmetric matrices

Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...

**0**

votes

**1**answer

130 views

### “locally” factoring subgroups of Lie groups

I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).
I start with a subgroup ...

**3**

votes

**4**answers

984 views

### Parametrization of O(3)

Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?

**0**

votes

**0**answers

261 views

### Orthogonal Projections in Lie Theory

I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...

**9**

votes

**2**answers

2k views

### Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...

**-1**

votes

**2**answers

598 views

### The lie algebra of the orthogonal group of an arbitrary space time metric

Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...

**4**

votes

**2**answers

458 views

### Abelianization of Lie groups

If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...