![(h) Let Hi and H2 be defined based on X1 and X2 using (1). Verify that Hi H2. (i) Find the projection of y = [4 2 0 1]T onto](http://img.homeworklib.com/images/982283c8-9422-4dfd-8783-56264a9b464d.png?x-oss-process=image/resize,w_560)
Please show all work in READ-ABLE way. Thank you so much in advance.
Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that for any y E Rn, Hy e V y and show that e E et e Hint: Use [Im(X)ke(XT)
(d) Show that I-H is symmetric and idempotent (hence an orthogonal projection matrix). Can you guess what subspace it projects onto? Since H is symmetric, it has a spectral decomposition of the form H-UAUT, where U is orthogonal and A is diagonal. (e) Show that A2- A. What does this tell you about the eigenvalues of H? Hy be the projection of y onto Im(X) and Pick any vector y є Rn and let y e (I - H)y the error (or residua
(f) Show that ll ll2llell2 Hint: Recall that llyllyy, expand and use properties of H and I - H. (g) Show that for any ує Rn, we have утНУ-IIHy112. É H a PSD matrix? The projection matrix H in (1) only depends on the Im(X) and not the particular basis used to describe it. Let us consider the following concrete examples: -2 0 Here Im(X) -Im(X2) (can you see why?)
(h) Let Hi and H2 be defined based on X1 and X2 using (1). Verify that Hi H2. (i) Find the projection of y = [4 2 0 1]T onto Im(X1) and compute llell2 (the squared norm of the residual). Can you explain what is going on?
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Please solve both parts of this question! I've stared at it for
a long time without knowing how to approach it.
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Need help with linear algebra problem!
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