Lecture 4: Eigenvalues and Eigenvectors
Description
Professor Strang begins this lecture talking about eigenvectors and eigenvalues and why they are useful. Then he moves to a discussion of symmetric matrices, in particular, positive definite matrices.
Summary
\(Ax =\) eigenvalue times \(x\)
\(A^2x =\) (eigenvalue)\(^2\) times \(x\)
Write other vectors as combinations of eigenvectors
Similar matrix \(B = M^{-1}AM\) has the same eigenvalues as \(A\)
Related section in textbook: I.6
Instructor: Prof. Gilbert Strang
Problems for Lecture 4
From textbook Section I.6
2. Compute the eigenvalues and eigenvectors of A and _A_−1. Check the trace!
A = \(\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]\) and _A_−1 = \(\left[\begin{array}{cc}-1/2&1\\1/2&0\\\end{array}\right]\) .
_A_−1 has the ____ eigenvectors as A. When A has eigenvalues _λ_1 and _λ_2, its inverse has eigenvalues ____.
11. **The eigenvalues of A equal the eigenvalues of A**T. This is because det(A − λI) equals det(_A_T − λI). That is true because ____. Show by an example that the eigenvectors of A and _A_T are not the same.
15. (a) Factor these two matrices into A = _X_Λ_X_−1:
A = \(\left[\begin{array}{cc}1&2\\0&3\\\end{array}\right]\) and A = \(\left[\begin{array}{cc}1&1\\3&3\\\end{array}\right]\) .
(b) If A = _X_Λ_X_−1 then _A_3 = ( )( )( ) and _A_−1 = ( )( )( ).
