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I have been preparing for a competitive exam and was solving some old papers. I came across this question that involves direct sum. I learned about it in textbooks, However I dont know how to approach the problem:

Question: Matrix M is given as follows,

M = [ 1 -1 2] [-1 1 2] [ 2 2 -2] 3*3 matrix

The magnitude of the product of the minimum and maximum eigenvalues of the matrix H given by,

Sum of{I^(1-m) ⊗ M ⊗ I^(8-m)} m running from 0 to 8 I is Identity matrix and I^(k) = I⊗I⊗I.....k times

⊗ is the direct product

I understand that expanding this series is tedious and time consuming. There must be another way, a property of direct product of some sorts. Please help me approach this problem

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    $A\otimes B$ usually is the [Kronecker product](https://en.wikipedia.org/wiki/Kronecker_product) of matrices. What is the direct product of matrices? – 2017-02-13
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    Hi, to my understanding both are same – 2017-02-13
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    Then the wikipedia page can help you how to expand this series easily. See how the Kronecker product with identity is computed, on both sides! – 2017-02-13
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    Thank you. I will work on that and get back. – 2017-02-14

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