Subsection 2.1.3 What you will learn¶
This week introduces two concepts that have theoretical and practical importance: unitary matrices and the Singular Value Decomposition (SVD).
Upon completion of this week, you should be able to
Determine whether vectors are orthogonal.
Compute the component of a vector in the direction of another vector.
Relate sets of orthogonal vectors to orthogonal and unitary matrices.
Connect unitary matrices to the changing of orthonormal basis.
Identify transformations that can be represented by unitary matrices.
Prove that multiplying with unitary matrices does not amplify relative error.
Use norms to quantify the conditioning of solving linear systems.
Prove and interpret the Singular Value Decomposition.
Link the Reduced Singular Value Decomposition to the rank of the matrix and determine the best rank-k approximation to a matrix.
Determine whether a matrix is close to being nonsingular by relating the Singular Value Decomposition to the condition number.