Let qeRn be a unit vector. Define the nxn matrix Hq = 2P, - Inxn where P, is as in the previous exercise and Inxn is the nxn identity matrix. a. Is H, an orthogonal projector? If not, which property of an orthogonal projector does He fail to satisfy? b. As a direct follow up to a calculation that you had to do in the previous subexercise, what is the inverse of H? Hint: Do not try to use Gauss-Jordan elimination to construct the inverse. See if you can find a matrix Hil from your calculations in (a) such that H; Hq = H,H1 = Inxn. c. One way to define an orthogonal matrix Q is that it is square and has orthonormal column vectors. Another equivalent definition is that Q-1 = QT. In light of your calculations from (a) and (b), is H, an orthogonal matrix? d. In the subexercises that follow, take dx=( i. Using GeoGebra, MATLAB, etc. or by hand, sketch the line spanned by q. Onto your sketch of this line, plot x and Hox. Connect the tips of these vectors with a straight line. ii. What geometrically does multiplying x by Hq do to x? Does this at all contradict the conclusion you arrived at in (a)? Remark: Hy is called a Householder reflector. These matrices are at the heart of a much more numerically stable way to compute the QR factorization of a matrix, known as the Householder algorithm. Let qeRn be a unit vector. Define the nxn matrix Hq = 2P, - Inxn where P, is as in the previous exercise and Inxn is the nxn identity matrix. a. Is H, an orthogonal projector? If not, which property of an orthogonal projector does He fail to satisfy? b. As a direct follow up to a calculation that you had to do in the previous subexercise, what is the inverse of H? Hint: Do not try to use Gauss-Jordan elimination to construct the inverse. See if you can find a matrix Hil from your calculations in (a) such that H; Hq = H,H1 = Inxn. c. One way to define an orthogonal matrix Q is that it is square and has orthonormal column vectors. Another equivalent definition is that Q-1 = QT. In light of your calculations from (a) and (b), is H, an orthogonal matrix? d. In the subexercises that follow, take dx=( i. Using GeoGebra, MATLAB, etc. or by hand, sketch the line spanned by q. Onto your sketch of this line, plot x and Hox. Connect the tips of these vectors with a straight line. ii. What geometrically does multiplying x by Hq do to x? Does this at all contradict the conclusion you arrived at in (a)? Remark: Hy is called a Householder reflector. These matrices are at the heart of a much more numerically stable way to compute the QR factorization of a matrix, known as the Householder algorithm