Answered: 0-2-1021 7. ( Compute the full SVD of… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.9: Properties Of Determinants

Problem 40E

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Explain and simplify

0-2-1021
7. (
Compute the full SVD of the following matrix A, THE rank-1 approximation
A(1), and the approximation error ||A - Â(1)||2, where || ||2 is the spectral norm; show
all work.
A =
223
3
2

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3. Let A = 112₁2 [45] 2-2 PCP ¹. Find an invertible matrix P and a matrix C of the form C b b (Z such that A 4. Find the line y = Bo+Bir that best fits the data {(1,1), (2, 1), (3,2), (4,4)} in the least-squares sense.
-1 1 -1 3.) Given the matrix A = -2 1 1 (a) Find the singular values o; of A. [Hint: first show that the characteristic polynomial of (AT A) is X³ – 13 X2 + 36 A.] (b) Find the rank 1 matrix A' that is closest to A (with respect to the Frobenius norm). Note: you do NOT need to find the whole SVD to do this. Find only those parts you need for A'. (c) Find both the Frobenius norm ||A||F and the 2-norm ||A||2 of A. Note: the matrix 2-norm is subordinate to the vector 2-norm.
Am.111.
Let 4 = [1 2] a. Give the characteristic equation of A and verify that A satisfies the characteristic equation. b. Confirm that (A − 31)(A − 1) = O, where I is the 2x2 identity matrix and O is the 2x2 zero matrix. How does this relate to part a?
0 ... Use the method of least squares to fit a straight line to the five data points. (х): -2 —1 0 1 2 (у): 4 3 3 1 -1 a) Construct Y and X matrices for the data. b) Find X'X and X'Y. c) Find the least squares estimates B = (X'X)-'X'Y. Hint: The inverse of a diagonal matrix 1 d11 d11 .. d22 0 1 d22 D = is D-1 0 0 . dnn 1 0 0 .. dnn Provide the fitted regression equation. Obtain a point estimate for the mean of Y when X = 1. Calculate the residual.
Do parts (a) (d) and (e) of the following: Consider the model y = B1 +x2B2 + x3ß3 + e and suppose that application of least squares to 20 observations on these variables yields the following results (cov (b) denotes the estimated covariance matrix): [0.96587] bị b2 b3 0.21812 0.019195 -0.050301 0.69914 cov(b) 0.019195 0.048526 –0.031223 1.7769 -0.050301 -–0.031223 0.037120 ở = 2.5193 R = 0.9466 %3D (a) Find the total variation, unexplained variation, and explained variation for this model. (b) Find 95% interval estimates for B2 and B3. (c) Use a t-test to test the hypothesis Ho:B2 >1 against the alternative Hj :B2< 1. (d) Use your answers in part (a) to test the joint hypothesis Họ:B2 = 0, B3 = 0. (e) Test the hypothesis Ho:2B2 = B3.
10 k Q11. Show that the matrix 0 1 is similar to an 0 0 1 orthogonal matrix if and only if k = 0. use both spatial/geometric intuition and eigenvalues/eigenspaces.
9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
Exercises 2.1. In each of the following cases, compute a basis matrix for the null space of the matrix A and express the points X; as x; = Pi +q; where p; is in the null space of A and q; is in the range space of A’. (ii) -2 A = (1 1 1 1), x = X2 = 6. -13 -2. 1. 75 4.
5
3.4.22 (a)
2.2. Let A be a square matrix with real entries, and let λ be its complex eigenvalue. Suppose v = (v₁, V2,...,Un) is a corresponding eigenvector, Av = Av. Prove that the X is an eigenvalue of A and Av=Xv. Here v is the complex conjugate of the vector V, V = (1, U2,..., Un).

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