Problem 2. (40 pts) Given the matrix A A =

Problem 2. (40 pts) Given the matrix A $A = \begin{bmatrix} 1 & 0 \ 0 & 1 \ 1 & 1 \end{bmatrix}$. 2.1 Perform the QR factorization of matrix A using the Gram-Schmidt process. That is, A = QR where Q is an orthogonal matrix and R is an upper triangular matrix. Show all your steps. 2.2 Find the singular value decomposition (SVD) of A. Show all your steps.

Transcript

00:01 Hi, today we are solving the question in which we are given with the matrix a is equals to 1 0 minus 1 and 0 1 1.

00:10 So here we need to find matrices in the u sigma b in the single value decomposition of a.

00:18 So here first of all finding the transpose of the given matrix.

00:26 So a transpose will be equal to 1 0 0 1 minus 1 and 1.

00:34 Now taking multiplying the matrix with its transpose.

00:39 So taking it as w is equals to 1 0 minus 1 0 1 1 multiplied to 1 0 0 1 minus 1 and 1.

00:52 On multiplication we get the matrix w as 1 0 minus 1 0 1 1 minus 1 1 and 2.

01:05 Now by using the ion value and ion vector of w.

01:10 So here we get ion value lambda 1 as 3 and v1 as minus 1 by 2 1 by 2 and 1.

01:21 Here done the ion value and ion vector before to make the calculation much easier and smaller.

01:31 Then lambda 2 is equals to 1 and v2 will be equal to 1 1 0 lambda 3 will be equal to 0 and v3 will be equal to 1 minus 1 and 1.

01:48 These are the ion values and ion vectors of the matrix w.

01:53 Now finding the square roots of the nonzero ion value.

01:57 So taking sigma 1 is equals to root 3 then sigma 2 is equals to 1.

02:07 The sigma matrix is a 0 matrix with sigma i on its diagonal.

02:12 So here we get the matrix sigma as root 3 0 0 1 0 0.

02:23 Similarly the columns of the matrix u are the normalized unit vectors.

02:29 So we get u is equals to minus root 6 by 6 minus root 2 by 2 then root 3 by 3 then root 6 by 6 then root 2 by 2 then minus root 3 by 3...

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