Let be an eigenvalue of an n n matrix A. Its geometric multiplicity is defined as dim Nul(A I). The goal of this problem is to show that the geometric multiplicity is less than or equal to the algebraic multiplicity. a) Set m = dim Nul(A I). Show that A is similar to a matrix of the form Im C O D , where Im is the n n identity matrix, C is some (n m) (n m) matrix, and D is some m m matrix. b) Show that the algebraic multiplicity of is bigger than or equal to m.

photo below https://imgur.com/a/I29wJXB

Problem 5. Let A be an eigenvalue of an n x n matrix A. Its geometric multiplicity is defined as dim Nul(A - AI). The goal of this problem is to show that the geometric multiplicity is less than or equal to the algebraic multiplicity. a) Set m = dim Nul(A - XI). Show that A is similar to a matrix of the form o D , where Im is the n x n identity matrix, C is some (n - m) x (n - m) matrix, and D is some m x m matrix. b) Show that the algebraic multiplicity of A is bigger than or equal to m