Learn · Matrix Theory and Applications
Lecture 3 of 76Week 1
Norms, distances, and the Cauchy–Schwarz inequality
Prof. Aditya Siripuram · IIT Hyderabad
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- Matrix Theory and Applications, IIT Hyderabad (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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All the lectures
- Vectors: definitions and operations
- Inner products and linear functions
- Norms, distances, and the Cauchy–Schwarz inequality
- Projection onto a vector
- Vector spaces and general inner products
- Norms beyond Euclidean
- Complexity of vector operations
- Matrices and their action
- Matrix products and transpose
- More on the transpose
- A battery of examples: structural matrix families
- A battery of examples: matrices from applications
- Block matrices, linear maps, and the Jacobian
- Systems and examples
- Existence and structure
- Example systems, stability
- Matrix maps: injectivity, surjectivity, inverses
- Null space and linear independence
- Subspaces and the image space
- Easy facts about linear independence
- Growing linearly independent sets
- Dimension and basis
- Rank and nullity Part_1
- Rank and nullity Part _2
- The rank–nullity theorem
- Rank, the transpose, and the inverse
- Examples: rank and nullity at work
- Orthogonal complements
- Orthonormal bases and the residual
- Gram–Schmidt and the QR decomposition
- Solving systems and finding bases with QR
- Projections and least squares
- Cost, and the trouble with Gram–Schmidt
- Recap: least squares
- Projections in general
- Least squares with constraints
- solving constrained least squares
- Regularization and gradient descent
- Optional - Solving augumented System
- Recursive Least squares - part1
- Recursive Least squares - part2
- Multi Objective Least squares
- Non linear Least squares
- Matrix Least squares
- (Optional) Estimation:Random variables as vectors
- Dimensionality Reduction and graph drawing
- Repeated matrix action and quadratic forms
- Eigenvectors and eigenvalues
- Diagonalizability
- The spectral theorem and rayleigh quotient
- Dimensionality Reduction and graph drawing in one dimension
- Recurrences through the eigenbasis
- The power method and the disc theorem
- Powers of stochastic Matrix - Part 1
- Powers of stochastic Matrix - Part 2
- Gradient descent on Least squares - Part 1
- Gradient descent on Least squares - Part 2
- The Trace
- Ky fan's theorem : maximizing over k directions
- Dimensionally reduction abd graph drawing in two dimensions
- Determinants and the characteristic polynomial
- Optional: Proof of the spectral theorem - Part 1
- Optional: Proof of the spectral theorem - Part 2
- Singular values via low rank approximation - Part 1
- Singular values via low rank approximation - Part 2
- Singular values via low rank approximation - Part 3
- The singular value decomposition - Part 1
- The singular value decomposition - Part 2
- The pseudo inverse
- Spectral norm and condition number
- Orthogonal Procustes
- Low rank approximation
- Sketching for least squares - Part 1
- Sketching for least squares - Part 2
- Sketching for least squares - Part 3
- Kaczmarz's method and stochastic gradient descent for least squares