CHAPTER 04 / 40
Matrix Decompositions
7 lessons · Explore, experiment and understand.
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Eigendecomposition and SVD power dimensionality reduction, recommender systems, low-rank attention, and the spectral analysis under every modern model.
- PCA
- recommender systems
- low-rank attention
- Step 18~20m3D · 125★Most cinematic⚡Challenge
Determinant and Trace
The determinant of a square matrix is a single scalar that captures the signed volume scaling factor of the linear map . If , the map triple…
Open lesson - Step 19~20m3D · 225★Most cinematic⚡Challenge
Eigenvalues and Eigenvectors
An eigenvector of a square matrix is a non-zero vector that stretches without rotating: for some scalar , the corresponding eigenvalue. Eige…
Open lesson - Step 20~15m15⚡Challenge
Cholesky Decomposition
The Cholesky decomposition factors a symmetric positive definite (SPD) matrix as , where is lower triangular with positive diagonal entries.…
Open lesson - Step 21~20m3D · 215★Most cinematic⚡Challenge
Eigendecomposition and Diagonalization
A square matrix is diagonalizable if it can be written as , where is diagonal and is invertible. The columns of are eigenvectors of and the…
Open lesson - Step 22~20m3D · 215★Most cinematic⚡Challenge
Singular Value Decomposition
The singular value decomposition (SVD) factors any matrix as , where and are orthogonal and is a (rectangular) diagonal matrix with non-nega…
Open lesson - Step 23~20m3D · 115★Most cinematic⚡Challenge
Matrix Approximation
Given a matrix of rank , the best rank- approximation (for ) is the matrix that minimizes among all rank- matrices. The answer is a celebrat…
Open lesson - Step 24~15m15⚡Challenge
Matrix Phylogeny
Let's step back and organize the decompositions you have met into a phylogeny — a family tree of matrix factorizations, each specialized for…
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