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Math 2331 { Linear Algebra - UH

DiagonalizationMath 2331 Linear DiagonalizationJiwen HeDepartment of Mathematics, University of jiwenhe/math2331 Jiwen He, University of HoustonMath 2331, Linear Algebra1 / DiagonalizationDiagonalization Theorem DiagonalizationDiagonalizationMatrix Powers: ExampleDiagonalizableDiagonalization TheoremDiagonalization: ExamplesJiwen He, University of HoustonMath 2331, Linear Algebra2 / DiagonalizationDiagonalization Theorem ExamplesDiagonalizationThe goal here is to develop a useful factorizationA=PDP 1,whenAisn n. We can use this to computeAkquickly for matrixDis adiagonalmatrix ( entries off the maindiagonal are all zeros).Powers of Diagonal MatrixDkis trivial to compute as the following example [5 00 4]. ComputeD2andD3. In general, what isDk,wherekis a positive integer?

a. For 1 k p, the dimension of the eigenspace for k is less than or equal to the multiplicity of the eigenvalue k. b. The matrix A is diagonalizable if and only if the sum of the dimensions of the distinct eigenspaces equals n, and this happens if and only if the dimension of the eigenspace for each k equals the multiplicity of k. c.

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