Learn · Linear Algebra for Machine Learning and Quantum Computing
Lecture 5 of 26Week 1
Least squares problems, reflections, rotations
Prof. Ranveer Singh · IIT Indore
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- From
- Linear Algebra for Machine Learning and Quantum Computing, IIT Indore (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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- Tweens profiles and older. How courses are placed
All the lectures
- Matrix product as linear combinations
- Linear dependence and bases
- Inner products and projections
- Projection onto subspaces and projection matrices
- Least squares problems, reflections, rotations
- Gram-Schmidt orthogonalization and QR decomposition
- Four fundamental subspaces and their orthogonality
- Matrix rank and the rank-nullity theorem
- Gaussian elimination, pivots, and computing rank
- Matrix inverses and rank tests for linear systems
- Least-squares linear regression and normal equations
- Determinants, elimination, and invertibility
- Eigenvalues, eigenvectors, and the characteristic equation
- Diagonalization using eigenvectors, matrix powers, and long-term behavior
- Symmetric matrices, spectral theorem, and orthogonal diagonalization
- Positive definite matrices, quadratic forms, Cholesky decomposition, and optimization
- Singular value decomposition: singular values, singular vectors, construction, and rank
- Principal component analysis: variance, covariance, projection, and dimensionality reduction
- PCA through SVD, low-rank approximation, image compression, and Eckart–Young theorem
- Random walks, Markov chains, stationary distributions, spectral mixing, and Monte Carlo simulation
- Clustering, K-Means, and Introduction to Spectral Clustering
- Mathematical Reasoning for Spectral Clustering
- Complex Linear Algebra, Qubits, and Measurement
- Single-Qubit Gates: Actions and Examples
- Multi-Qubit States, Gates, and Density Matrices
- Separable States and Entanglement