Numerical Linear Algebra : Unitary and Triangular Matrices and Krylov Matrix
Not what you're looking for?
1. Let A = QR be the factorization of a A into the product of a unitary matrix and a trianglular matrix. Suppose that the cloumns of A are linearly independent. Show that |rkk| is the distance from the k-th column of A to the linear space spanned by the first k-1 columns of A.
2. Let A Є C^(mxm) and b Є C^m be abitrary. Show that anyy x Є Kn is equal to p(A)b for some polynomial p of degree < n- 1.
Note: Kn is the mxn Krylov matrix
(See attached file for full problem description)
Purchase this Solution
Solution Summary
Unitary and Triangular Matrices and Krylov Matrix are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Proof:
QR factorization of A is defined as follows:
A = [Q1 | Q2] , or
Ie, QR factorization of A splits the problem Ax = b into two subproblems:
Qy = b and Rx = y
Which are solvable, since y = Q-1b and R is the upper triangular matrix.
Also given that the columns of matrix A are linearly independent. Let's define Householder matrix as follows:
W = I - 2u.uT , with ||u|| = 1.
This matrix W is orthonormal:
Ie., WTW = (I-2u.uT)T.( (I-2u.uT) = 1- 4u.uT + 4u(uTu)uT = 0
And we now choose the vector u so that multiplying by W zeros out the columns of A as shown below.
Let a be a ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Probability Quiz
Some questions on probability
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.