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Lecture Notes (23)

# Lecture Notes (23) - MA 3280 Lecture 23 More on...

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MA 3280 Lecture 23 - More on Eigenvectors Wednesday, April 2, 2014. Objectives: Discuss the diagonalization of a matrix. There is something called a diagonalization for a matrix. This comes easily from the eigenvalues and eigenvectors. For the matrix (1) A = b 1 - 1 2 4 B . We got the eigenvalues λ = 2 , 3, and eigenvectors of the form (2) b - a a B and b - a 2 a B . We can diagonalize A by putting any particular eigenvectors for each eigenvalue, for example (3) b 1 - 1 B and b 1 - 2 B , as columns in a matrix (4) P = b 1 1 - 1 - 2 B , ±nding the inverse (5) P - 1 = 1 - 2 - ( - 1) b - 2 - 1 1 1 B = b 2 1 - 1 - 1 B . We get D = P - 1 AP , or (6) b 2 1 - 1 - 1 B b 1 - 1 2 4 B b 1 1 - 1 - 2 B = b 4 2 - 3 - 3 B b 1 1 - 1 - 2 B = b 2 0 0 3 B . Linear maps We can de±ne a function from an R m to an R n by multiplying vectors by a matrix. For example, we can de±ne the function L : R 2 R 2 by (7) L pb x 1 x 2 B P = b 2 0 0 3 B b x 1 x 2 B = b 2 x 1 3 x 2 B , which takes vectors in R 2 and maps them to vectors in R 2 . This function is an example of something we’ll call a linear map .

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Lecture Notes (23) - MA 3280 Lecture 23 More on...

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