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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 2

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 2

Identify the set \(M(n)\) of all \(n\times n\) matrices with real entries with \(\mathbb {R}^{n^2}\) as usual. Show that the set \[ O(n)=\{M\in M(n)\mid MM^t=I\} \] is a submanifold of \(\mathbb {R}^{n^2}\) (here \(M^t\) denotes the transpose of \(M\) and \(I\) is the identity matrix).

Raw LaTeX

Identify the set \(M(n)\) of all \(n\times n\) matrices with real entries with
\(\mathbb{R}^{n^2}\) as usual. Show that the set
\[
  O(n)=\{M\in M(n)\mid MM^t=I\}
\]
is a submanifold of \(\mathbb{R}^{n^2}\) (here \(M^t\) denotes the transpose of \(M\) and
\(I\) is the identity matrix).

Topic Tags

Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds