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I'm a graduate student in a field of science where we frequently need to optimize a matrix in $U(n)$ or $SO(n)$ (henceforth $SO(n)$ for concreteness) to get an "optimal" orthonormal basis before doing a more complicated second optimization. We perform our first optimization by varying over the Lie algebra. Almost always, the Lie algebra is too large a variational space because the basis is only determined up to orthogonal transformations of special subsets. Accordingly, we eliminate the generators of the unwanted orthogonal transformations. This is usually done without comment, but the one source I've found that attempts to justify that we don't lose any cosets this way uses a matrix decomposition. The hypotheses of the decomposition are not stated, so this is my best attempt to construct them:

Consider $SO(n)$. Given a composition of $n$, let $K$ be the special orthogonal group "applied to" that composition via direct product, e.g., $SO(100)$ has decomposition ${80, 14, 6},$ and $K = SO(80) otimes SO(14) otimes SO(6)$. $K$ is a Lie subgroup of $G$, with Lie subalgebra $mathfrak{k}$. Therefore as vector spaces, $mathfrak{g} = mathfrak{k} oplus mathfrak{p}$. We then have $$G = K exp(mathfrak{p}) = exp(mathfrak{p}) K$$

The original paper references no proof of the decomposition. This decomposition is especially relevant to a current research project of mine, so I would like to give a proper citation. This seems very similar to Cartan decomposition, but in the case where you have more than two invariant subspaces, the condition $[mathfrak{p}, mathfrak{p}] subset mathfrak{k}$ fails, and Cartan decomposition cannot be used directly. While you could do a sequence of Cartan decompositions, this would give a product of exponentials, which isn't what my field uses, and different sequences give exponentials of different subsets of the parent Lie algebra.

This brings me to the following questions:

1. Is this matrix decomposition valid as stated?


2. If it is valid, is there a reference for it? An ideal reference is clearly applicable to my specific case, proves the theorem, and avoids specialized Lie theoretic machinery.


3. If it is not valid, is there some substitute to justify eliminating the "redundant" parameters?

In case these are relevant:

• While the problem should also be expressible in the language of Grassmann manifolds, I'm much less familiar with that language.


• My own background in Lie theory is an undergraduate course through Stillwell's Naive Lie Theory plus odds and ends I've picked up while trying to identify this matrix decomposition.


• Lie theory is not a standard tool in my field. In particular, differential geometry is unheard of.