# 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.

Question
Similarity
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.

2021-03-10
Step 1
Let, the rank of (A−3I) is 1. A matrix is diagonalizable if for each eigen value $$\displaystyle\lambda$$, the rank r of the matrix
(A−3I)=n-the multiplicity of $$\displaystyle\lambda$$ , where n= size of the matrix
Step 2
A $$\displaystyle{n}\times{n}$$ matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible $$\displaystyle{n}\times{n}$$ matrix c and a diagonal matrix D such that $$\displaystyle{A}={C}{D}{C}^{{−{1}}}$$
Example:
$$\displaystyle{A}={\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}={I}_{{3}}{\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}{{I}_{{3}}^{{-{{1}}}}}$$
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.

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