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.

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Tasneem Almond · Accepted Answer

Step 1 Let, the rank of (A−3I) is 1. A matrix is diagonalizable if for each eigen value λ, the rank r of the matrix (A−3I)=n-the multiplicity of λ , 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=CDC−1 Example: A=[100050006]=I3[100050006]I3−1 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.

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.

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