Lecture: The Singular Value Decomposition (SVD)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when they are working through the SVD derivation and geometric intuition and get stuck on specific steps in the lecture.
Recommended by 0 people across 0 different months, 2017 to 2021.
What it is
A 44-minute university lecture from the AMATH 301 course on the singular value decomposition, covering its mathematical foundations and the geometric picture of a matrix mapping a sphere to an ellipsoid. The lecturer is Nathan Kutz, and it is billed as perhaps the most important concept in the course.
What people said
Cites MrPmosh · 2017-07 · Mathematics Stack Exchange · on “AMATH 301 SVD Lecture Doubt”
In This lecture SVD Lecture [this video] @20:58 when the lecturer is writing down the transformation generalization: Why is the vector matrix a square matrix? (n x n notation) What about if I have 3 vectors in a 2-dim system? wouldn't it be a 2x3 matrix? Is it because we're 'representing' "n-spheres" in orthonormal systems? (requiring a n x n vector matrix to be defined?) (My Linear Algebra is rusty)
Cites An Ignorant Wanderer · 2020-07 · Mathematics Stack Exchange · on “Why does a matrix multiplication transform a circle to an ellipse?”
In this Youtube video [this video] , Nathan Kutz explains that if you "hit" a circle with a matrix multiplication, it becomes an ellipse. In the same video, he mentions that a matrix multiplication is linear, so it will stretch any two different vectors by the same amount (minute 9:13 to 9:45). Given the visual that he provided (which I've attached below), I don't see why the circle would go to an ellipse if you're stretching (and rotating) v1 and v2 by the same amount. It seems to me that you would just get a circle with a larger radius. [LINK] I've come across a similar question to mine [LINK] , but I don't think it answers my question.
Cites SoraHeart · 2021-09 · Mathematics Stack Exchange · on “Singular Value Decomposition Solving the U & V”
I am following this video [this video] on YouTube. I have a question on 41 minute mark. When solving for $V$ , $A^t A V = V \Sigma^2$ , have been rearrange to $$Ax = \lambda x$$ Why can $V \Sigma ^2$ become $\Sigma ^ 2 V$ ? I thought matrix multiplication is non-commutative. [LINK]