MATH 323 Lecture 26

From Notes
Jump to navigation Jump to search

« previous | "Thursday", November 4, 2012 | next »

Theorem 6.3.1

If are distinct eigenvalues of an matrix with corresponding eigenvectors , then are linearly independent.


is said to be diagonalizable if there is a nonsingular matrix with

where is a diagonal matrix.

Therefore by .

Theorem 6.3.2

An matrix is diagonalizable iff has linearly independent eigenvectors.


  1. If is diagonalizable and , then diagonal entries of are eigenvalues of .
  2. , but is not unique.
  3. If eigenvalues are distinct, then is diagonalizable
  4. If they are not distinct, then may or may not be diagonalizable
  5. If , then .


Matrix Exponential

is a very important function.

We similarily define for a matrix to be

If A is diagonalizable, then , where is given by


Application: Differential Equation

Solution to differential equation is .

Similarly, the system of differential equations has the solution .

Final Exam Review

5 problems and an 11-point bonus problem

  • Linear transformations (Null Space, Range, Basis)
  • Orthogonality
  • Orthogonalization (Gram-Schmidt process, factorization)
  • Eigenvalues and Eigenvectors
  • Least Squares Problems

Note on Factoring Cubics

For a cubic polynomial , if for , divides :