Question

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0 Orthogonally diagonalize the following matrix 3 7 A= 0 5 0 7 0 3

Answer 1

Given 3 wot 0 5 7 subtract t from the diagonal entries of matrix 13-X 7 5-> 7 3-) Find determinant 13-X 5-) 3-) 0 7 o O - 200 =0. = -13+1 12²+rod-2 The roots are: 2, 310,12=5,13=-4, These are eigenvalues. Next, find eigenvectors. D.d=10; 3-10 0 7 o 5-10 7 o 3-10 o O -5 2 o Reduce this matrix

U O Now, solve the matoix equation si O 100 If we take &z=t, then v=t, 120, V3=t. ..Ve 11-18 t. let tal :) IT 2-5 오 2 o --2 The obtained matrix after subtracting das. Reduce this matrix, we get oo ооо Now, solve the matrix equation i o 16:30-la If we take Va=t, then W, =0, V₂ =t, Vz=0. V= 1:1-03 (: t let t=1 2 III) a=-4: 13-(-4) ct o tort Oo o o 7 o 7 o 3-1-4) Reduce this matrix c 0 1 o OO Now, solve the matrix equation si D: 818 - It we take &z=t, then u, = -t, 2 = 0, V3=t 13] t v= I let tal t

Therefore, are 10,5,-4 eigenvaluer Eigenvectors [:]» (), 6) invertible matrix composed of eigenvectos p is an o O o ! pt = adjoint (P) la 0 ya IPI - O 1/2 1/2 Then PAP 0 = o Va o کم 16:10 13 o - o 5 no 5 O diagonal matlin o 10 Therefore, pt AP =D Hence A is Othogonally diagonalized.