# When is a rank one matrix diagonalizable? Justify your answer. What is one choice of the diagonalizing similarity? What happens when it is not diagonalizable? Justify your answer.

When is a rank one matrix diagonalizable? Justify your answer. What is one choice of the diagonalizing similarity? What happens when it is not diagonalizable? Justify your answer.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Tasneem Almond

Step 1
Let, the rank of $\left(A-3I\right)$ is 1. A matrix is diagonalizable if for each eigen value $\lambda$, the rank r of the matrix
$\left(A-3I\right)=n$-the multiplicity of $\lambda$ , where $n=$ size of the matrix
Step 2
A $n×n$ matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible $n×n$ matrix c and a diagonal matrix D such that $A=CD{C}^{-1}$
Example:
$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 5& 0\\ 0& 0& 6\end{array}\right]={I}_{3}\left[\begin{array}{ccc}1& 0& 0\\ 0& 5& 0\\ 0& 0& 6\end{array}\right]{I}_{3}^{-1}$
Ay diagonal matrix A is diagonalizable as it is similar to itself.
Step 3
If the matrix is not diagonalizable, the one has to find a matrix with same properties consisting of eigen values on the leading diagonal and either ones or zeros on the super diagonal.