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Eigenvalues, eigenvectors and applications

Linear transformations on planeEigen valuesMarkov MatricesEigenvalues, eigenvectors and applicationsDr. D. SukumarDepartment of MathematicsIndian Institute of Technology HyderabadRecent Trends in Applied Sciences with Engineering ApplicationsJune 27-29, 2013 Department of Applied ScienceGovernment Engineering College,Kozhikode, KeralaDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesMaps which preserveOriginlines passing through originparallelograms with one corner as originDr.

Eigenvalues, eigenvectors and applications Dr. D. Sukumar Department of Mathematics Indian Institute of Technology Hyderabad Recent Trends in Applied Sciences with Engineering Applications June 27-29, 2013 Department of Applied Science Government Engineering College,Kozhikode, Kerala Dr. D. Sukumar (IITH) Eigenvalues

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Transcription of Eigenvalues, eigenvectors and applications

1 Linear transformations on planeEigen valuesMarkov MatricesEigenvalues, eigenvectors and applicationsDr. D. SukumarDepartment of MathematicsIndian Institute of Technology HyderabadRecent Trends in Applied Sciences with Engineering ApplicationsJune 27-29, 2013 Department of Applied ScienceGovernment Engineering College,Kozhikode, KeralaDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesMaps which preserveOriginlines passing through originparallelograms with one corner as originDr.

2 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesOutline1 Linear transformations on planeTypical ExamplesProperties2 Eigen valuesEigen value and eigen vector3 Markov MatricesFormationInterpretationPropertie sDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesRotation(0 110)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesRotation(0 110)-112-112xyxy-112-112Dr.

3 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesReflection(100 1)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesReflection(100 1)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesExpansion(2 00 2)Compression(1/2001/2)-112-112xyxy-112- 112xy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesExpansion(2 00 2)Compression(1/2001/2)-112-112xyxy-112- 112xy-112-112Dr.

4 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesExpansion(2 00 2)Compression(1/2001/2)-112-112xyxy-112- 112xy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesMulti-scaling or Stretching(2 00 3)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesMulti-scaling or Stretching(2 00 3)-112-112xyxy-112-112Dr.

5 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesProjection(1 00 0)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesProjection(1 00 0)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesShear transformation(1 10 1)-112-112xyxy-112-112Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesShear transformation(1 10 1)-112-112xyxy-112-112Dr.

6 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesOutline1 Linear transformations on planeTypical ExamplesProperties2 Eigen valuesEigen value and eigen vector3 Markov MatricesFormationInterpretationPropertie sDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesPropertiesAreaEigen vectorsEigen valuesDeterminantDiagonalizableDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesPropertiesAreaEigen vectorsEigen valuesDeterminantDiagonalizableDr.

7 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesPropertiesAreaEigen vectorsEigen valuesDeterminantDiagonalizableDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesPropertiesAreaEigen vectorsEigen valuesDeterminantDiagonalizableDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesPropertiesAreaEigen vectorsEigen valuesDeterminantDiagonalizableDr.

8 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesTypical ExamplesPropertiesTable of propertiesMapAreaFixed DirScale in FDDetDiagonableEigenvectorEigenvalueRota tion1 NONO1 NOReflection1x-axis,y-axis1,-1-1 YesExpansion4x-axis,y-axis2, 24 YesCompression1/4x-axis,y-axis1/2,1/21/4 YesMulti-scaling6x-axis,y-axis2,36 YesProjection0x-axis,y-axis1,00 YesShear1x-axis11 NOTable: PropertiesDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorProblemBig ProblemGetting a common opinion from individual opinionFrom individual preference to common preferencePurposeShowing all steps of this process using linear algebraMainly using eigenvalues and eigenvectorsDr.

9 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorProblemBig ProblemGetting a common opinion from individual opinionFrom individual preference to common preferencePurposeShowing all steps of this process using linear algebraMainly using eigenvalues and eigenvectorsDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorProblemBig ProblemGetting a common opinion from individual opinionFrom individual preference to common preferencePurposeShowing all steps of this process using linear algebraMainly using eigenvalues and eigenvectorsDr.

10 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorProblemBig ProblemGetting a common opinion from individual opinionFrom individual preference to common preferencePurposeShowing all steps of this process using linear algebraMainly using eigenvalues and eigenvectorsDr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorOutline1 Linear transformations on planeTypical ExamplesProperties2 Eigen valuesEigen value and eigen vector3 Markov MatricesFormationInterpretationPropertie sDr.

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