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].

5.3 Diagonalization DiagonalizationTheoremExamples Diagonalization: Example Example Diagonalize the following matrix, if possible. A = 2 4 2 4 6 0 2 2 0 0 4 3 5: Since this matrix is triangular, the eigenvalues are 1 = 2 and 2 = 4. By solving (A I)x = 0 for each eigenvalue, we would nd the following: 1 = 2 : v 1 = 2 4 1 0 0 3 5, 2 = 4 : v 2 = 2 ...

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