1. Let A = gallery(5). Answer questions in Problem 10.17 for this matrix. Problem 10.17 Is ill-conditioned? Find the eigenvalues of Perturb by a random matrix whose values do not exceed −5 Does...

1. Let A = gallery(5). Answer questions

in Problem 10.17 for this matrix.

Problem 10.17

Is
ill-conditioned?

Find the eigenvalues of

Perturb

by a random matrix

whose values do not exceed

⁻⁵

Does it appear the eigenvalues of

are ill-conditioned

Find eigenvectors for

using the command

Perturb the elements of A by
*
and compute the eigenvectors. What conclusion can you make?

2. A magic square of order

is an

matrix with entries

²

such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. MATLAB will create a magic square with the command
.

What is the largest eigenvalue of magic(n) for

In general, what is the largest eigenvalue of a magic square of order n? Prove it.

Look at the Perron-Frobenius theorem, Theorem 5.6 in this book.

What is the largest singular value of magic(n) for

In general, what is the largest singular value of a magic square of order