Math 2331 { Linear Algebra

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?Jiwen He, University of HoustonMath 2331, Linear Algebra3 / DiagonalizationDiagonalization Theorem ExamplesDiagonalization (cont.)Solution:D2=[5 00 4][5 00 4]=[00]D3=D2D=[520042][5 00 4]=[00]and in general,Dk=[5k004k]Jiwen He, University of HoustonMath 2331, Linear Algebra4 / DiagonalizationDiagonalization Theorem ExamplesMatrix Powers: ExampleExampleLetA=[6 123].

Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 18. 5.3 Diagonalization DiagonalizationTheoremExamples Diagonalization (cont.) Solution: D2 = 5 0 0 4 5 0 0 4 = 0 0 D3 = D2D = 52 0 0 42 5 0 0 4 = 0 0 and in general, Dk = 5k 0 0 4k

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