Tag: arrowhead matrices
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Wrangling quartics, II
In the last post on this topic, we saw that when optimizing the objective$$ {1 \over 2 n^2 } \|X^T Z^T Z X – C\|_F^2 + {\la \over 2 m^2}\|Z – I\|_F^2,$$ any solution $Z$ is symmetric and satisfies $${2 \over n^2} \left( XX^T Z^2 XX^T – X C X^T\right) + {\la \over m^2} I…
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Inverting arrowhead matrices.
I need to invert a matrix of the form $$ M = I + S^2 H S^2,$$ where $H$ is a symmetric matrix, and $S^2$ is diagonal. The elements of $S^2$ drop off very quickly, so what remains of $H$ are its first column and first row, scaled by $S_{1}^2 S^2$. The result is that…