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As many readers are no doubt aware, the title of this blog comes from the famous book Geometry and the Imagination by Hilbert and Cohn-Vossen (based on lectures given by Hilbert). One of the first things discussed in that book is the geometry of conics, especially in two and three dimensions. An ellipsoid is a certain kind of (real) quadric surface, i.e. a surface in $\mathbb{R}^n$ defined by a single quadratic equation of the co-ordinates. It may also be defined as the image of the unit $(n-1)$-dimensional sphere under an affine self-map of $\mathbb{R}^n$. After composing with a translation, one may imagine an ellipsoid centered at the origin, and think of it as the image of the unit sphere under a linear automorphism of $\mathbb{R}^n$ — i.e. transformation by a nonsingular matrix $M$.

A (generic) ellipsoid has $n$ axes; in dimension three, these are the “major axis”, the “minor axis” and the “mean axis”. Distance to the origin is a Morse function on a generic ellipsoid; the symmetry of an ellipsoid under the antipodal map means that critical points occur in antipodal pairs. There are a pair of critical points of each index between $0$ and $n$. There is a gradient flow line of this Morse function between each pair of critical points whose index differs by $1$, and the union of these flowlines are the ($2$-dimensional) ellipse obtained by intersecting the ellipsoid with the plane spanned by the pair of axes in question. This shows that these axes are mutually perpendicular.

One may use this geometric picture to “see” the $KAK$ decomposition of $\text{GL}(n,\mathbb{R})$ as follows, where $K$ denotes the orthogonal subgroup $\text{O}(n,\mathbb{R})$, and $A$ denotes the subgroup of diagonal matrices with positive entries. Let $M$ be a linear map of $\mathbb{R}^n$, and let $E$ be the ellipsoid which is the image of the unit sphere under $M$. Let $\xi_i$ be the axes of $E$ of index $i$. There is a unique orthogonal matrix $O$ taking the $\xi_i$ to the co-ordinate axes. There is a unique diagonal matrix $D$ taking $O(E)$ to the round sphere. Hence the composition $ODM$ is orthogonal, and we can express $M$ as a product of an orthogonal matrix, a diagonal matrix, and another orthogonal matrix.

One can use ellipsoids to visualize another less standard matrix decomposition as follows. For simplicity we concentrate on the case of dimension $3$. The minor and mean axis span a plane $\pi$ which intersects the ellipsoid in the “smallest” possible ellipse. Rotate this plane by keeping the mean axis fixed, and tilting the minor axis towards the major axis. At some unique point one obtains a plane $\pi'$ that intersects the ellipsoid in a round circle. One may shear the ellipsoid, keeping this plane fixed, into an ellipsoid of rotation. This describes a way to factorize $M$ as a product of a shear, a diagonal matrix with two equal eigenvalues, and a rotation.

Question: What is the generalization of the “shear, dilate, rotate” factorization in higher dimensions?

Question: Is there a way to see the Iwasawa ($KAN$) decomposition geometrically, by using ellipsoids?