Youtor
Visualize Spectral Decomposition SEE Matrix Chapter 2
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=mhy-ZKSARxI
A video illustrating the underlying elegant visual interpretation of Spectral Decomposition. • Chapters: • 0:00 Chapter 1 Summary • 1:23 Symmetric Matrix ? • 1:44 Property of Transpose • 3:27 Matrix Decomposition • 4:58 Eigen Vectors and Eigen Values • 8:07 Strong Property of Symmetric Matrix • 9:36 Spectral Decomposition • 12:03 Visualization • 14:13 appreciation • • Video Sins: • 1. regarding the eigenvectors of symmetric, it is correct to say the eigen vectors are orthogonal if the matrix is full rank. However, the formal definition is there always exist a orthornormal basis which contains the eigen vectors of the symmetric matrix, for details refer to • 25. Symmetric Matrices and Positive D... , where professor Strang explains the case with eigen vectors with eigen value of 0. • 2. 00:50: a diagonal matrix is also a symmetric matrix, and its visual interpretation is very straightforward. So it’s definitely not correct to say every symmetric matrix produces a visually complicated transformation. But normally when we address a matrix as a symmetric, we often mean “symmetric and non-diagonal”, if the matrix is diagonal, we would just call it diagonal matrix. • 3. 2:46, the transpose of orthogonal matrix is also a orthogonal matrix. I didn’t mention this fact. • 4. 7:42, when the eigen value is 0, it means the vector is in the null-space, pretty important fact, which I didn’t mention. • 5. 14:20: the visual interpretation of matrix decomposition doesn’t apply to every type of matrix decomposition. Most matrix decomposition are often used for speeding up computational purposes, for example the LU decomposition. • 6. Order of eigen vector could be swap, so there are many alternative decomposition of symmetric matrices. • This video wouldn’t be possible without the inspiration of the legendary 3b1b : • / 3blue1brown • and the animation software - Manim, which he wrote: • / 3blue1brown • and the Manim Community: • https://docs.manim.community/en/stabl...
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