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Vijay posted an Question

July 21, 2020 • 14:01 pm

30 points

IIT JAM
Mathematics (MA)

1 1 2 the largest eigen value of the matrix 1-2 5 is 2 5 -3

1 1 2 The largest eigenvalue of the matrix 1-2 5 is 2 5 -3 Please show me the solution by solving the eigenvalue the hard way.

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Piyush pachauri

PFA. Greshgorin theorem Application.
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There should be max out of bracket in the Last
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|Lamda| <= max{3,5,1}. what does this statement tell? I know that it is the sum of each modulus of the elements in the first row I understand that but what it's used for?
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it implies any Eigen value of this matrix cannot be greater than 5
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that is the max value Eigen value can take.
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but it looks like any Eigenvalue of that matrix can not be greater than 3.
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Deepak singh 1

see attached
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Piyush pachauri

PFA. When matrix entries are positive , greshgorin theorem is easy way to go.
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Ask if you have any doubt.
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I appreciate your handwriting plus your effort. To be honest, I knew about this trick already but I wanted it in such a way where we'd have to solve the cubic equation and from the roots, we'd choose the largest that's how I wanted it to be solved and the theorem sounds good but to be true, I didn't just get the theorem. if you can then consider taking some time to explain to me the theorem.
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No problem ,I will explain this theorem.
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