Matrix Lie groups and smooth subgroups

The invertible real n-by-n matrices form an open subset of the vector space of all matrices, since nonzero determinant is an open condition. Matrix multiplication is polynomial in the entries. Inversion is smooth there because the adjugate formula divides polynomial entries by a nonvanishing determinant. Consequently GL-n is a Lie group of dimension n squared. Neither the singular matrices nor the ambient vector space with its usual matrix multiplication form a group of the same kind.

A subgroup inherits a candidate multiplication, but it still needs an appropriate manifold structure. One useful construction is a regular level set of smooth matrix constraints. Another is the closed subgroup theorem: a subgroup closed in a finite-dimensional Lie group has a unique embedded Lie subgroup structure compatible with the ambient one. This is stronger than merely observing that a parametrization is smooth, and its hypothesis is relative closedness in the ambient group rather than necessarily closedness in all matrices.

The orthogonal group is defined by transpose-A times A equal to the identity. The constraint map takes matrices into symmetric matrices. Its derivative at the identity sends an increment X to X plus its transpose, a surjection onto the symmetric matrices. The implicit function theorem therefore gives the expected local dimension n times n minus one divided by two. Translation in the group carries the same manifold description to every orthogonal matrix. This argument counts independent equations rather than subtracting all n squared entries as if they were independent.

Taking determinants shows that an orthogonal matrix has determinant plus or minus one. The special orthogonal subgroup selects the plus sign and has the same local dimension, because this discrete choice does not add a new local equation. In dimension two the plus component consists of rotations, while a reflection belongs to the other component. Matrix constraints therefore reveal both continuous degrees of freedom and discrete information that a tangent space alone cannot remember.

A closed matrix subgroup is closed relative to GL-n. For example, the positive scalars form a closed and open subgroup of the multiplicative group of nonzero real scalars, even though they are not closed in the whole real line. A dense one-parameter subgroup of a torus supplies a different phenomenon: its natural immersed Lie group topology can be finer than the subset topology inherited from the torus. An injective smooth group map need not be an embedding.

When checking a proposed matrix symmetry family, verify closure under multiplication, identity, and inverse before studying its geometry. A surface of matrices can look smooth while failing group closure, and a group can have multiple components with the same local equations. Later, tangent Lie algebras simplify calculations near the identity; they do not replace this initial topological bookkeeping. Distinguishing intrinsic parameters, ambient entries, and group components makes dimension claims both useful and verifiable.