Hello, I am sending this message to have the answer to this exercise.

for question a) especially and b) if you have the time (not necessary). Here is the statement:

it's bilinear algebra at undergraduate level.

This exercise is related to the "isometries and similarities" part of my bilinear algebra course.

I haven't really understood bilinear algebra, I sometimes apply formulas but I don't know if they are the right ones.

Exercise 2.3. Let M be an orthogonal matrix in Mn (R), meaning that it satisfies *MM = In. a Show that if x is a real eigenvalue of M, then r E {-1, 1}. b/ Show that if zo is a complex non-real eigenvalue of M, then its conjugate Zo is also an eigenvalue of M (consider X as an eigenvector of M in C" corre- sponding to zo, and show that the vector X with conjugate coordinates is an eigenvector for Zo)