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Prove that the special orthogonal group of matrices is compact and connected.

I am trying to prove that the special orthogonal group

SO_2(\mathbb{R})

is connected and compact

SO(n)=

{

n \times n

matrices

A \in O(n): det A=1

}

O(n)=

{

n \times n

matrices A \in GL(n)}

----------

Compactness
-----------

To show compactness, I was thinking of using the Heine-Borel criterion that a subset

V \subset \mathbb{R^{n}}

is compact

\iff

V is closed and bounded.

SO_2(\mathbb{R})

is closed since it is the preimage of

1

under

det

Is it bounded?

----------
Connectedness

To show

SO_2(\mathbb{R})

is connected I must show that it does *not* split into two disjoint, non-empty open subsets. I was thinking of doing a proof by contradiction: suppose

SO_2(\mathbb{R})

splits into two disjoint, non-empty open subsets...I think the 'open' condition fails. But I cannot think of a formal proof though

Would really appreciate your help. Thanks

(edited 8 years ago)

Reply 1

joostan

Original post by number23

I am trying to prove that the special orthogonal group

SO_2(\mathbb{R})

is connected and compact

SO(n)=

{

n \times n

matrices

A \in O(n): det A=1

}

O(n)=

{

n \times n

matrices

A \in GL(n)}

----------

Compactness
-----------

To show compactness, I was thinking of using the Heine-Borel criterion that a subset

V \subset \mathbb{R^{n}}

is compact

\iff

V is closed and bounded.

SO_2(\mathbb{R})

is closed since it is the preimage of

1

under

det

Is it bounded?

----------
Connectedness

To show

SO_2(\mathbb{R})

is connected I must show that it does *not* split into two disjoint, non-empty open subsets. I was thinking of doing a proof by contradiction: suppose

SO_2(\mathbb{R})

splits into two disjoint, non-empty open subsets...I think the 'open' condition fails. But I cannot think of a formal proof though

Would really appreciate your help. Thanks

To show it's bounded, try using the fact that if

M \in SO(2)

then

MM^T=I

.

As for connectedness, I'm not entirely sure but you might get somewhere identifying

SO(2)

\mathbb{R}^4

or failing that a quick google search suggests you could go for path connectivity through the identity.

Reply 2

Gregorius

Original post by number23

I am trying to prove that the special orthogonal group

SO_2(\mathbb{R})

is connected and compact

As @joostan has pointed out, boundedness follows by looking at the condition for membership of the orthogonal group. Connectedness is usually proved by going via path connectedness. To show the latter one usually reaches for the fact that elements of special orthogonal groups can be represented as compositions of an even number of reflections. Then an argument connecting reflections to the identity finishes it off.

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Prove that the special orthogonal group of matrices is compact and connected.

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