031 Review Part 3, Problem 3

Let  $A={\begin{bmatrix}5&1\\0&5\end{bmatrix}}.$

(a) Find a basis for the eigenspace(s) of  $A.$

(b) Is the matrix  $A$  diagonalizable? Explain.

Foundations:
Recall:
1. The eigenvalues of a triangular matrix are the entries on the diagonal.
2. By the Diagonalization Theorem, an  $n\times n$  matrix  $A$  is diagonalizable
if and only if  $A$  has  $n$  linearly independent eigenvectors.

Solution:

(a)

Step 1:
Since  $A$  is a triangular matrix, the eigenvalues are the entries on the diagonal.
Hence, the only eigenvalue of  $A$  is  $5.$

Step 2:
Now, to find a basis for the eigenspace corresponding to  $5,$  we need to solve  $(A-5I){\vec {x}}={\vec {0}}.$
To do this, we use row reduction. Thus, we get
${\begin{array}{rcl}\displaystyle {A-5I}&=&\displaystyle {{\begin{bmatrix}5&1\\0&5\end{bmatrix}}-{\begin{bmatrix}5&0\\0&5\end{bmatrix}}}\\&&\\&\sim &\displaystyle {{\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}\end{array}}$
Solving this system, we see  $x_{1}$  is a free variable and  $x_{2}=0.$
Therefore, a basis for this eigenspace is
${\bigg \{}{\begin{bmatrix}1\\0\end{bmatrix}}{\bigg \}}.$

(b)

Step 1:
From part (a), we know that  $A$  only has one linearly independent eigenvector.

Step 2:
By the Diagonalization Theorem,  $A$  must have  $2$  linearly independent eigenvectors to be diagonalizable.
Hence,  $A$  is not diagonalizable.

Final Answer:
(a)     The only eigenvalue of  $A$  is  $5$  and the corresponding eigenspace has basis  ${\bigg \{}{\begin{bmatrix}1\\0\end{bmatrix}}{\bigg \}}.$
(b)     $A$  is not diagonalizable.

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