References:
- Multivariate Statistics Lecture notes - by Shahin Tavokoli
- Choleski decompostion - blog post - notes
- Some very nice linear algebra blog posts
Spectral Decomposition and SVD
Singular Value Decomposition (SVD)
Any real $n \times p$ matrix $X$ admits the following decomposition:
\[\begin{align*} X = ULV^T \end{align*}\]Where $U$ is an $n \times n$ orthogonal matrix, $L$ is an $n \times p$ rectangular diagonal matrix with non-negative entries (singular values), and $V$ is a $p \times p$ orthogonal matrix.
This is saying that:
“Every matrix is diagonal provided one uses the proper bases for the domain and range spaces.”
And the geometric interpretation of this is that the linear transformation represented by any matrix is given by: a rotation, then scaling of the axis by the singular vlaues, and then rotating the axis again. Figure 4 in this blog post visualises this.
This also helps to explain to me why we take such an interest in eigenvalues/singular values, as they tells us by how much a linear transformation distort the spaces. This then helps to explain why matrices with zero valued eigenvalues are not invertible, as they collapse the space. In addition, why orthogonal matrices have all eigenvalues equal to 1, as they do not distort the space.
Spectral Decomposition
Let $A$ be a symmetric matrix, then $A$ admits the following decomposition:
\[\begin{align*} A = EDE^T \end{align*}\]Where $E$ is the orthogonal matrix with columns being the eigenvectors of $A$. And $D$ is a diagonal matrix with eigenvalues of $A$.
This has the same geometric interpretation but now the second rotation is the same as the first. This video provides a visualisation.
Relationship between the two
https://gregorygundersen.com/blog/2018/12/20/svd-proof/#1-gram-matrices-as-positive-semi-definite
http://theanalysisofdata.com/probability/C_5.html#:~:text=The%20following%20corollary%20shows%20that,are%20the%20eigenvectors%20of%20A.
Cholesky decomposition
- Cholesky decomposition exists $\iff$ $M$ is symmetric and positive definite.