Guide for Week 7
Math 408
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Reading Assignment:
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Finish reading Chapter 4 in the book, and read Chapter 5, Sections 1-3.
Homework Assignment:
Vocabulary Words
- Optimization of Quadratic functions
- quadratic functions
- the relationship between linear least squares and quadratic functions
- symmetric matrices
- why do symmetric matrices have only real eigenvalues?
- orthogonal matrices
- diagonal matrices
- eigenvalue decomposition for symmetric matrices
- positive definite/semi-definite matrices
- the Rayleigh-Ritz Theorem
- the existence and uniqueness theorem for quadratic optimization (unconstrained)
- the existence and uniqueness theorem for quadratic optimization with linear constraints
- the optimal value in a quadratic optimizations problem when an optimal solution does not exist
- H-conjugate vectors
- the conjugate gradient algorithm
- Elements of Multivariable Calculus
- continuity
- open, closed, bounded, and compact sets
- directional derivatives and partial derivatives
- derivatives and their representations as matrices of partial derivatives
- the little-o notation
- the gradient and the Hessian matrix
- first-order and second-roder expansion of a function
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Key Concepts:
- Optimization of Quadratic functions
- the relationship between linear least squares and quadratic functions
- eigenvalue decomposition for symmetric matrics
- the Rayleigh-Ritz Theorem
- positive definite/semi-definite matrices
- the existence and uniqueness theorem for quadratic optimization
- the existence and uniqueness theorem for quadratic optimization with linear constraints
- the conjugate gradient algorithm
- Elements of Multivariable Calculus
- first- and second-order expansions and approximations
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Skills to Master:
- solving a quadratic optimization problem
- computing derivatives and Taylor approximations
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Quiz:
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The quiz will consist of 2 questions.
The first question will be related to vocabulary
words listed above.
The second question will be computational
and will be similar to the problems in
the homework set.