A First Course in Linear Algebra · Problem C22

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · diagonalization

Problem

Consider the matrix

A

below. Find the eigenvalues of

A

using a calculator and use these to construct the characteristic polynomial of

A

p_A(x)

. State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for

A

by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is

A

diagonalizable? If not, explain why. If so, find a diagonal matrix

D

that is similar to

A

A = \begin{bmatrix} 19 & 25 & 30 & 5 \\ -23 & -30 & -35 & -5 \\ 7 & 9 & 10 & 1 \\ -3 & -4 & -5 & -1 \end{bmatrix}