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4. Let X € R"xm. Show that x™x >0.

4. Let X € R"xm. Show that x™x >0.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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1. Let A = |0 0. What is the range space of A? What is the rank of A? |0 2 What is the null space of A and the null space of AT. [1 2] 2. Obtain the eigendecomposition of A = 2 1 Use the eigendecomposi- tion to obtain Tr A and det A. Find ||A||2 and ||A|| F. 3. Let a = |ị x2 13 T4 15]. Find det(I + aa"). Find ||a||?. 4. Let X € R"xm Show that XxTx>0. 5. Let y = f(x) = t, where A E Rmxm is non-singular, 6, x, C e R", d e R. Find the inverse_function, i.e., find f-1 such that x = f-'(y). is non-singular. [A b] Assume that Q = 6. Show that for A symmetric inf, A : Amin (A). 7. Let f(x) = LA. Compute Vf(x). Show that Vf(x) = 0 if and only if x is an eigenvector of A.
Transcribed Image Text:1. Let A = |0 0. What is the range space of A? What is the rank of A? |0 2 What is the null space of A and the null space of AT. [1 2] 2. Obtain the eigendecomposition of A = 2 1 Use the eigendecomposi- tion to obtain Tr A and det A. Find ||A||2 and ||A|| F. 3. Let a = |ị x2 13 T4 15]. Find det(I + aa"). Find ||a||?. 4. Let X € R"xm Show that XxTx>0. 5. Let y = f(x) = t, where A E Rmxm is non-singular, 6, x, C e R", d e R. Find the inverse_function, i.e., find f-1 such that x = f-'(y). is non-singular. [A b] Assume that Q = 6. Show that for A symmetric inf, A : Amin (A). 7. Let f(x) = LA. Compute Vf(x). Show that Vf(x) = 0 if and only if x is an eigenvector of A.
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