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Aditi Agrawal posted an Question

June 03, 2020 • 14:21 pm

30 points

IIT JAM
Physics (PH)

Plewse explain this also. here if lambda 1 is negative then is it necessary that its value be -1.if yes then why

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Abhishek singh Best Answer

okay, so your doubt is about lambda 1, right? now listen, as the given matrix is orthogonal matrix, it's Eigen values would have modulus equal to one. two Eigen values lambda2 and lambda3 are complex conjugate of each other. then lambda1 could be either 1 or -1 ,right! so as Rubi said, product of Eigen values is equal to determinant. that's why product of lambdas is equal to -1. now as written in your notes, lambda2*lambda3 = a^2+b^2, will is always positive. hence in lambda1 has to be negative to make the total product(or determinant) equal to -1(negative).
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if you have doubt in any particular step of my explanation, do ask that particular doubt.
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i want to ask that for one eigen valve we will have only one eigen vector
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how does this result comes that for n free variables we will have n eigen vectors
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what is free variable? free variable means you can choose them according to you. so if you have two free variable, you can choose as max two independent vectors. which means two eigenvalues.
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ok thanks
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Ruby negi

hope you are getting...
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i have asked my doubt in your above comments reply
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Ruby negi

in the question, lamda1=-1. now you have to find lamda2,3.. also determinant and trace is given that is -1 and 2/3.. use you can make 2equation - first trace-(lamda1)=lamda2+lamda3. and second determinant/lamda1=lamda2*lamda3... using these two you can easily find lamda2 ,lamda3 (because u have 2variable and 2equations)...
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Ruby negi

dear,note product of eigen values is equal to determinant of the matrix and sum of eigen values is equal to trace of matrix... for if you have given one eigen values,then using these two conditions(trace and determinant) u can easily find other two eigen values...
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but here we have supposed the value of lambda 1 as -1.if we do not suppose this value then what would be the solution
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i understood this but again i want to ask that it is not given in the question that value of lambda 1 is -1 this we have supposed from the cubic equation. but if we dont do this then what would be the solution
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Dhairya sharma

dear, determinant is defined as the product of the alll the eigen values.........and trace is the sum of all the eigen values of the matrix. since we find out the product of the remaining eigen value which is positive. now remaining is -ve so the product will be negative. try
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dear see attached.....u will get an idea
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