Lecture 60 of 67Week 11
Spectral Theorem for Hermitian Matrices
Prof. Arbind Kumar Lal · IIT Kanpur
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- Linear Algebra, IIT Kanpur (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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All the lectures
- Notations, Motivation and Definition
- Matrix: Examples, Transpose and Addition
- Matrix Multiplication
- Matrix Product Recalled
- Matrix Product Continued
- Inverse of a Matrix
- Introduction to System of Linear Equations
- Some Initial Results on Linear Systems
- Row Echelon Form (REF)
- LU Decomposition - Simplest Form
- Elementary Matrices
- Row Reduced Echelon Form (RREF)
- Row Reduced Echelon Form (RREF) Continued
- RREF and Inverse
- Rank of a matrix
- Solution Set of a System of Linear Equations
- System of n Linear Equations in n Unknowns
- Determinant
- Permutations and the Inverse of a Matrix
- Inverse and the Cramer's Rule
- Vector Spaces
- Vector Subspaces and Linear Span
- Linear Combination, Linear Independence and Dependence
- Basic Results on Linear Independence
- Results on Linear Independence Continued...
- Basis of a Finite Dimensional Vector Space
- Fundamental Spaces associated with a Matrix
- Rank - Nullity Theorem
- Fundamental Theorem of Linear Algebra
- Definition and Examples of Linear Transformations
- Results on Linear Transformations
- Rank-Nullity Theorem and Applications
- Isomorphism of Vector Spaces
- Ordered Basis of a Finite Dimensional Vector Space
- Ordered Basis Continued
- Matrix of a Linear transformation
- Matrix of a Linear transformation Continued...
- Matrix of Linear Transformations Continued...
- Similarity of Matrices
- Inner Product Space
- Inner Product Continued
- Cauchy Schwartz Inequality
- Projection on a Vector
- Results on Orthogonality
- Results on Orthogonality.
- Gram-Schmidt Orthonormalization Process
- Orthogonal Projections
- Gram-Schmidt Process: Applications
- Examples and Applications on QR-decomposition
- Recapitulate ideas on Inner Product Spaces
- Motivation on Eigenvalues and Eigenvectors
- Examples and Introduction to Eigenvalues and Eigenvectors
- Results on Eigenvalues and Eigenvectors
- Results on Eigenvalues and Eigenvectors
- Results on Eigenvalues and Eigenvectors.
- Diagonalizability
- Diagonalizability Continued...
- Schur's Unitary Triangularization (SUT)
- Applications of Schur's Unitary Triangularization
- Spectral Theorem for Hermitian Matrices
- Cayley Hamilton Theorem
- Quadratic Forms
- Sylvester's Law of Inertia
- Applications of Quadratic Forms to Analytic Geometry
- Examples of Conics and Quartics
- Singular Value Decomposition (SVD)
- Live Session 20-10-2020