Eigenvalues, eigenvectors and applications - IIT Hyderabad

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. 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.

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

3 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. 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.

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

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

6 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. 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.

7 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. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorEigen valueLetAa be square values ofAare solutions or roots ofdet(A I) = xor(A I)x= 0,for a non-zero vectorxthen is an eigenvalue ofAandxis an eigenvector corresponding to the eigenvalue .Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorEigen valueLetAa be square values ofAare solutions or roots ofdet(A I) = xor(A I)x= 0,for a non-zero vectorxthen is an eigenvalue ofAandxis an eigenvector corresponding to the eigenvalue.

8 Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorEigen valueLetAa be square values ofAare solutions or roots ofdet(A I) = xor(A I)x= 0,for a non-zero vectorxthen is an eigenvalue ofAandxis an eigenvector corresponding to the eigenvalue .Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorEigen valueLetAa be square values ofAare solutions or roots ofdet(A I) = xor(A I)x= 0,for a non-zero vectorxthen is an eigenvalue ofAandxis an eigenvector corresponding to the eigenvalue .Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr.

9 D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].

10 Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr. D. Sukumar (IITH)EigenvaluesLinear transformations on planeEigen valuesMarkov MatricesEigen value and eigen vectorExampleConsider the matrixA=[2 53 0].Example (Eigen value)A I=[2 53 0] [1 00 1]=[2 53 ]det(A I) = (2 )( ) (3 5) = 2 2 15 The roots of the polynomial are the eigen values: -3 and (When = 3)[2 ( 3)53 ( 3)](x1x2)=(00)Eigen vector( 11)Example (When = 5)[2 553 5](x1x2)=(00)Eigen vector(53)Dr.