Yog A yog m × n matrix, ces ATA thiab AAT muaj tib yam nonzero eigenvalues … Yog li ntawd Ax yog ib qho eigenvector ntawm AAT sib raug rau eigenvalue λ. Ib qho kev sib cav sib cav tuaj yeem siv los qhia tias txhua qhov tsis yog xoom eigenvalue ntawm AAT yog eigenvalue ntawm ATA, yog li ua tiav cov pov thawj.
Puas eigenvalues ntawm AAT thiab ATA tib yam?
Cov matrices AAT thiab ATA muaj tib yam nonzero eigenvalues. Tshooj 6.5 tau qhia tias eigenvectors ntawm cov symmetric matrices yog orthogonal.
Puas yog ATA tib yam li AAT?
Vim AAT thiab ATA yog qhov sib luag tiag tiag, lawv tuaj yeem ua rau diagonalized nrog orthogonal matrices. Nws ua raws los ntawm cov nqe lus dhau los (txij li geometric & algebraic multiplicities coincide) uas AAT thiab ATA muaj tib yam eigenvalues.
Puas yog ATA muaj qhov txawv txav?
Nyob. Piv txwv li, yog A= 1 2 3 2 4 −1 3 −1 5 , ces tus cwj pwm sib npaug det(A − λI)=−25 − 15λ + 10λ2 − λ3=0 tsis muaj lub hauv paus rov qab. Li no tag nrho eigenvalues ntawm A yog qhov txawv thiab A yog diagonalizable. 3.35 Rau txhua qhov matrix A, AtA ib txwm diagonalizable.
Yuav ua li cas sib txawv eigenvectors muaj tib eigenvalue?
Ob qhov txawv Eigenvectors sib raug zoo ib yam Eigenvalue ib txwm nyob ntawm. Ob qhov sib txawv Eigenvectors sib haum rau tib Eigenvalue yog ib txwm nyob ntawm linearly.
