question1.dviQuestion 1 : The elements of a n× n matrix A is given by,Aij = i ∗ jwhere, the indices...

Question

question1.dviQuestion 1 : The elements of a n× n matrix A is given by,Aij = i ∗ jwhere, the indices i and j runs from 1 to n. Find the eigenvalues of the matrixand their degeneracies.1

David · Accepted Answer

answer1.dvi
Answer 1 : Since,
Aij = i ∗ j
Hence, A11 = 1 ∗ 1 = 1, A12 = 1 ∗ 2 = 2, A13 = 1 ∗ 3 = 3 and so on.
The explicit form of A is given by,
A =












1 2 3 4 .......... n− 1 n
2 4 6 8 .......... 2n− 2 2n
3 6 9 12 .......... 3n− 3 3n
4 8 12 16 .......... 4n− 4 4n
.. .. .. .. .......... ..... 5n
.. .. .. .. .......... ..... ...
.. .. .. .. .......... ..... ...
.. .. .. .. .......... ..... ...
n 2n 3n 4n .......... n2 − 1 n2












Now, the eigenvalue equation for A reads,












1 2 3 4 .......... n− 1 n
2 4 6 8 .......... 2n− 2 2n
3 6 9 12 .......... 3n− 3 3n
4 8 12 16 .......... 4n− 4 4n
..

question1.dvi Question 1 : The elements of a n× n matrix A is given by, Aij = i ∗ j where, the indices i and j runs from 1 to n. Find the eigenvalues of the matrix and their degeneracies. 1

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