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Numerical Linear Algebra : Unitary and Triangular Matrices and Krylov Matrix

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 &#1028; C^(mxm) and b &#1028; C^m be abitrary. Show that anyy x &#1028; Kn is equal to p(A)b for some polynomial p of degree < n- 1.
Note: Kn is the mxn Krylov matrix

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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 ...

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