Iwasawa decomposition

In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

Then the Iwasawa decomposition of ${\displaystyle {\mathfrak {g}}_{0}}$ is

and the Iwasawa decomposition of G is

The dimension of A (or equivalently of ${\displaystyle {\mathfrak {a}}_{0}}$) is equal to the real rank of G.

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