Let's diagonalise (if we can)!

Algebra Level 3

True or False?

If a real n × n n\times n matrix A A satisfies the equation ( A − 2 I n ) ( A − 3 I n ) = 0 (A-2I_n)(A-3I_n)=0 , then there must exist an invertible real n × n n\times n matrix S S such that S − 1 A S S^{-1}AS is diagonal.

(from a recent test in linear algebra)

No Yes Impossible to determine

Otto Bretscher by Otto Bretscher , 65, USA

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1 solution

A A is diagonalizable iff its minimal polynomial is a product of non- repeated linear factors. And this is the case: ( A − 2 I n ) ( A − 3 I n ) = 0 (A - 2I_n) (A - 3I_n) = 0 gives us 3 possibilities fot its minimal polynomial:

a) A = 2 I n A = 2I_n

b) A = 3 I n A = 3I_n or

c) Its minimal polynomial is ( A − 2 I n ) ( A − 3 I n ) = 0 (A - 2I_n) (A - 3I_n) = 0 .Then there exists an invertible matrix S S such that S − 1 A S S^{-1}AS is a diagonal matrix with entries 2 and 3 in the diagonal.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

True, but this is a fact that people may not learn in a first course on linear algebra. Can you elaborate a bit?

Otto Bretscher - 2 years, 6 months ago

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I have elaborated a bit, is it better?

Guillermo Templado - 2 years, 6 months ago

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Yes, thank you. I don't know about Spain, but in the US we don't usually teach the kids about minimal polynomials in a first course on linear algebra, and very few take a second course. In a first course, we are happy if they understand that "a matrix is diagonalisable iff there exists a basis consisting of eigenvectors."

I may post a more elementary proof at some point if I get around to it.

Otto Bretscher - 2 years, 6 months ago

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