Solution Found!
Suppose A is a symmetric matrix with eigenvalues 2 and 5. If the vectors 1 1 1 and 1 1 0
Chapter 6, Problem 2(choose chapter or problem)
Suppose A is a symmetric matrix with eigenvalues 2 and 5. If the vectors 1 1 1 and 1 1 0 span the 5-eigenspace, what is A 1 1 2 ? Give your reasoning.
Questions & Answers
QUESTION:
Suppose A is a symmetric matrix with eigenvalues 2 and 5. If the vectors 1 1 1 and 1 1 0 span the 5-eigenspace, what is A 1 1 2 ? Give your reasoning.
ANSWER:Step 1 of 3
Suppose A is a symmetric matrix with eigenvalues 2 and 5 respectively.
The vectors 
To find
From the given hypothesis, it is clear that the eigenvalue 5 is of multiplicity 2 and the eigenvalue 2 is of multiplicity 1.
Since A is symmetric, by the spectral theorem, all eigenvalues of A are real and there exists an orthogonal matrix Q such that 
The diagonal elements of D are the eigenvalues of A. Therefore,
