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Let A E Mnxn (R) and assume that every nonzero element of R" is an eigenvector for A (all corresponding to real eigenvalues). Prove that there exists some u E R such that A = µ· In. (Hint: It may be helpful to consider the standard basis and some simple linear combinations.)

Let A E Mnxn (R) and assume that every nonzero element of R" is an eigenvector for A (all corresponding to real eigenvalues). Prove that there exists some u E R such that A = µ· In. (Hint: It may be helpful to consider the standard basis and some simple linear combinations.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 36EQ: Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has two...
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4. Let A E Mnxn(R) and assume that every nonzero element of R" is an eigenvector for A (all corresponding to real eigenvalues). Prove that there exists some u E R such that A = µ· In. (Hint: It may be helpful to consider the standard basis and some simple linear combinations.)
Transcribed Image Text:4. Let A E Mnxn(R) and assume that every nonzero element of R" is an eigenvector for A (all corresponding to real eigenvalues). Prove that there exists some u E R such that A = µ· In. (Hint: It may be helpful to consider the standard basis and some simple linear combinations.)
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