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nhigham.com | ||
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tiao.io
Suppose we're given a positive semidefinite (PSD) matrix $\mathbf{A} \in \mathbb{R}^{N \times N}$ to which we wish to update by some low-rank matrix $\mathbf{U} \mathbf{U}^\top \in \mathbb{R}^{N \times N}$, $$\mathbf{B} \triangleq \mathbf{A} + \mathbf{U} \mathbf{U}^\top,$$ where the update factor matrix $\mathbf{U} \in \mathbb{R}^{N \times M}$. To be more precise, the low-rank update is rank-$M$ for some $M \ll N$. What is the best way to calculate the Cholesky decomposition of $\mathbf{B}$? Given ...... |
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nickhar.wordpress.com
1. Low-rank approximation of matrices Let $latex {A}&fg=000000$ be an arbitrary $latex {n \times m}&fg=000000$ matrix. We assume $latex {n \leq m}&fg=000000$. We consider the problem of approximating $latex {A}&fg=000000$ by a low-rank matrix. For example, we could seek to find a rank $latex {s}&fg=000000$ matrix $latex {B}&fg=000000$ minimizing $latex { \lVert A - B... |
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stephenmalina.com
Matrix Potpourri # As part of reviewing Linear Algebra for my Machine Learning class, I've noticed there's a bunch of matrix terminology that I didn't encounter during my proof-based self-study of LA from Linear Algebra Done Right. This post is mostly intended to consolidate my own understanding and to act as a reference to future me, but if it also helps others in a similar position, that's even better! |
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entangledlogs.com
To visualize quaternions in the fanciest way, visit Ben eater, Quaternion. Euler angles suffer from a problem of gimbal lock. When rotating around a 3-perpendicular axis in euclidean space, if either two of these axes align i.e becomes parallel, it causes gimbal lock. Once the object is locked, the object will lose one degree of freedom for rotation. This video provides an intuitive explanation of the problem. Pitfalls When converting the Euler angle to a quaternion, it will lose some information. |
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