In this exercise, you will work with a QR factorization of an mn matrix. We will proceed in the way that is chosen by MATLAB, not by the author of our textbook.
Exercise 4 (4 points) In this exercise, you will work with a QR factorization of an mn matrix. we will proceed in the way that is chosen by MATLAB, not by the author of our textbook. Difficulty Moderate An mn matrix A can be presented as a product of a unitary (or orthogonal) mm matrix Q and an upper-triangular mxn matrix R, that is, A- Q R -1 Theory: a square mxm matrix Q is called unitary (or orthogonal) if Q 0, or equivalently, Q* is A QAQ. It is known that similar matrices have the same sets of eigenvalues - eye(m). It can be shown that the matrix A R*Q is similar to A, that For certain square matrices, such as, symmetrical, tridiagonal, Hessenberg matrices, the process of decomposing A into a product Q*R and interchanging the factors to create a matrix A,-R*Q, then, repeating the process with the matrix A, (instead of A) to create the matrix A, R and so on, produces a sequence of similar matrices A, A, A,.. that converges to a matrix, which is close to an upper triangular matrix, and its diagonal entries approximate the eigenvalues of A Note: QR is a powerful algorithm for finding the eigenvalues of certain matrices. **Create a function in MATLAB function [] guer (A m, n]-size (A); Part I: Generate the first factorization by using the MATLAB in-built function Q,R] qr (A) Output Q and R (do not put a semicolon after the command IQ,R]-qr (A). *Then, your function has to verify that you did get the factorization, that is, check if the condition below holds: closetozeroroundoff (A-R)0 If yes, output the message disp ('the product of Q and R forms a QR decomposition of A' If not, output something like disp ('No, it cannot be true!') The code for closetozeroroundoff is here function B-closetozeroroundoff (A) [m, n]-size (A) for i=1:m Exercise 4 (4 points) In this exercise, you will work with a QR factorization of an mn matrix. we will proceed in the way that is chosen by MATLAB, not by the author of our textbook. Difficulty Moderate An mn matrix A can be presented as a product of a unitary (or orthogonal) mm matrix Q and an upper-triangular mxn matrix R, that is, A- Q R -1 Theory: a square mxm matrix Q is called unitary (or orthogonal) if Q 0, or equivalently, Q* is A QAQ. It is known that similar matrices have the same sets of eigenvalues - eye(m). It can be shown that the matrix A R*Q is similar to A, that For certain square matrices, such as, symmetrical, tridiagonal, Hessenberg matrices, the process of decomposing A into a product Q*R and interchanging the factors to create a matrix A,-R*Q, then, repeating the process with the matrix A, (instead of A) to create the matrix A, R and so on, produces a sequence of similar matrices A, A, A,.. that converges to a matrix, which is close to an upper triangular matrix, and its diagonal entries approximate the eigenvalues of A Note: QR is a powerful algorithm for finding the eigenvalues of certain matrices. **Create a function in MATLAB function [] guer (A m, n]-size (A); Part I: Generate the first factorization by using the MATLAB in-built function Q,R] qr (A) Output Q and R (do not put a semicolon after the command IQ,R]-qr (A). *Then, your function has to verify that you did get the factorization, that is, check if the condition below holds: closetozeroroundoff (A-R)0 If yes, output the message disp ('the product of Q and R forms a QR decomposition of A' If not, output something like disp ('No, it cannot be true!') The code for closetozeroroundoff is here function B-closetozeroroundoff (A) [m, n]-size (A) for i=1:m
