# Any $2\times 2$ real matrix is orthogonal similar to a diagonal matrix.

Prove by direct matrix calculation that any $2\times 2$ real matrix is orthogonal similar to a diagonal matrix.

I got absent for a week in my class so i have less knowledge of it.. So all I could think about using based on what I have read is... $P^{T}AP = D$ , $D$ a diagonal matrix I just wanted some hints on how do I start with the proof. Should I generalize a 2 x 2 matrix given its properties and show that there exists D for all matrices? Thanks a lot for those who would reply.

Edit: I saw there was a theorem stating a real symmetric matrix is orthogonally diagonalizable... so.. the columns are eigenvectors then by G-S Orthogonalization process... I could find P and then its inverse and the diagonal... Is this structure of proof correct??

• I am assuming by "any $2\times 2$ real matrix is orthogonal similar to a diagonal matrix" that you mean "any $2\times 2$ orthogonal real matrix is similar to a real matrix"? – Math1000 Dec 6 '15 at 12:57
• $P^{T}.P=I$ That is orthogonal not $P^{-1}$ – Archis Welankar Dec 6 '15 at 13:12
• hmmm... I edited it I thought I made a mistake... thanks for clarifying. :) – Zurchi Dec 6 '15 at 13:17
• Please make titles informative and objective. – Pedro Tamaroff Dec 6 '15 at 13:20
• @Pedro It's my first time creating an account so spare me for my several mistakes XD... I'll be checking on these on my future posts. Thanks a lot!! – Zurchi Dec 6 '15 at 13:28