Problem 5 Consider the tollowing matrix -5 2-1 3 0 o1 2 a. Compute the eigenvalu

Question

Problem 5 Consider the tollowing matrix -5 2-1 3 0 o1 2 a. Compute the eigenvalues by hand. (Hint: Use the transpose AT to compute the eigenvalues using the method discussed in class. This eases computation). How do the eigenvalues compare to the diagonal entries of the matrix? Note, this property always holds true for an upper triangular matrix such as A (and a lower triangular matrix such as AT) b. For any repeated eigenvalues, find the eigenvectors by hand. Are the eigenvectors linearlyin- dependent? The number of times the eigenvalue is repeated is its algebraic multiplicity. What is the algebraic multiplicity for repeated eigenvalues? The geometric multiplicity is given by the max imum number of linearly independent associated eigenvectors. What is the geometric multiplicity for repeated eigenvalues? c. Consider the matrix B=| 0 What is the algebraic multiplicity and geometric multiplicity for any repeated eigenvalues? You may use Matlab. Is the matrix B diagonalizable?

Explanation / Answer

The given matrix is a upper triangular matrix. This means that the values below the principal diagonal are zero. For such matrices. The eigen values are the the values at the diagonal.

So for this matrix, we have dagonal values = {-5, 1, 1, 3}. And these are the eigen values.

Thanks.

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