Juan Hewlett

2021-12-15

Quick way to check if a matrix is diagonalizable.
Is there any quick way to check whether a matrix is diagonalizable or not?
In exam if a question is asked like "Which of the following matrix is diagonalizable?" and four options are given then how can one check it quickly? I hope my question makes sense.

usaho4w

Firstly make sure you are aware of the conditions of Diagonalize matrix.
In a multiple choice setting as you described the worst case scenario would be for you to diagonalize each one and see if its

Mary Goodson

One nice characterization is this: A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F.
So first, you can find the characteristic polynomial. If the characteristic polynomial itself is a product of linear factors over F, then you are lucky, no extra work needed, the matrix is diagonalizable.
If not, then use the fact that minimal polynomial divides the characteristic polynomial, to find the minimal polynomial. (This may not be easy, depending on degree of characteristic polynomial)

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