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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 .
An matrix is diagonalizable iff has linearly independent eigenvectors.
- If is diagonalizable and , then diagonal entries of are eigenvalues of .
- , but is not unique.
- If eigenvalues are distinct, then is diagonalizable
- If they are not distinct, then may or may not be diagonalizable
- If , then .
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)
- Orthogonalization (Gram-Schmidt process, factorization)
- Eigenvalues and Eigenvectors
- Least Squares Problems
Note on Factoring Cubics
For a cubic polynomial , if for , divides :