Diagonalization Revisted Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com.

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Presentation transcript:

Diagonalization Revisted Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com

A is diagonalizable if there exists an invertible matrix P such that P −1 AP = D where D is a diagonal matrix. Diagonalization has many important applications It allows one to convert a more complicated problem into a simpler problem. Example: Calculating A k when A is diagonalizable.

3 3

I.e., we are assuming A is diagonalizable since implies More diagonalization background:

Check answer:

To diagonalize a matrix A: Step 1: Find eigenvalues: Solve the equation: det (A –  = 0 for  Step 2: For each eigenvalue, find its corresponding eigenvectors by solving the homogeneous system of equations: (A –  x = 0 for x. Case 3a.) IF the geometric multiplicity is LESS then the algebraic multiplicity for at least ONE eigenvalue of A, then A is NOT diagonalizable. (Cannot find square matrix P). Matrix defective = NOT diagonalizable.

Case 3b.) A is diagonalizable if and only if geometric multiplicity = algebraic multiplicity for ALL the eigenvalues of A. Use the eigenvalues of A to construct the diagonal matrix D Use the basis of the corresponding eigenspaces for the corresponding columns of P. (NOTE: P is a SQUARE matrix). NOTE: ORDER MATTERS.

Step 1: Find eigenvalues: Solve the equation: det (A – I) = 0 for  For more complicated example, see video 4: Eigenvalue/Eigenvector Example & video 5: Diagonalization

characteristic equation: = -3 : algebraic multiplicity = geometric multiplicity = dimension of eigenspace = = 5 : algebraic multiplicity geometric multiplicity dimension of eigenspace 1 ≤ geometric multiplicity ≤ algebraic multiplicity

characteristic equation: = -3 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 = 5 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 1 ≤ geometric multiplicity ≤ algebraic multiplicity Matrix is not defective.

characteristic equation: = -3 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 = 5 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 1 ≤ geometric multiplicity ≤ algebraic multiplicity Matrix is not defective. Thus A is diagonalizable

characteristic equation: = -3 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 = 5 : algebraic multiplicity = 1 geometric multiplicity = 1 dimension of eigenspace = 1 1 ≤ geometric multiplicity ≤ algebraic multiplicity Matrix is not defective. Thus A is diagonalizable

Nul(A + 3  = eigenspace corresponding to eigenvalue = -3 of A Find eigenvectors to create P

Basis for eigenspace corresponding to = -3:

Nul(A - 5  = eigenspace corresponding to eigenvalue = 5 of A Find eigenvectors to create P Basis for eigenspace corresponding to = 5:

Basis for eigenspace corresponding to = -3: Basis for eigenspace corresponding to = 5:

Basis for eigenspace corresponding to = -3: Basis for eigenspace corresponding to = 5:

Note we want to be invertible. Note P is invertible if and only if the columns of P are linearly independent. We get this for FREE!!!!!

Note: You can easily check your answer.

Diagonalize Note there are many correct answers.

Diagonalize Note there are many correct answers. ORDER MATTERS!!!