In the following chapter we will describe how the

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Unformatted text preview: · 0 0 1 −2 · · ··· ··· ··· ··· ··· 0 0 · · −2 . We take k = n and m = (1/n), so that k/m = n2 . We can use the following Mathematica program to find the eigenvalues of the n × n matrix A, when n = 6: n = 6; m = Table[Max[2-Abs[i-j],0], {i,n-1} ,{j,n-1} ]; p = m - 4 IdentityMatrix[n-1]; 59 a = n∧2 p eval = Eigenvalues[N[a]] Since = 2 if j = i, 2 − |i − j | = 1 if j = i ± 1 ≤0 otherwise, the first line of the program generates 21 1 2 M = 0 1 · · 00 the (n − 1) × (n − 1)-matrix 0 ··· 0 1 ··· 0 2 ··· · . · ··· · · ··· 2 The next two lines generate the matrices P and A = n2 P . Finally, the last line finds the eigenvalues of A. If you run this program, you will note that all the eigenvalues are negative and that Mathematica provides the eigenvalues in increasing order. We can also modify the program to find the eigenvalues of the matrix when n = 12, n = 26, or n = 60 by simply replacing 6 in the top line with the new value of n. We can further...
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