For two square matrices, we say A is similar to B if there is a nonsingular third matrix X so that AX=XB. This makes it so that \(\mathbf{A}=\mathbf{X}\mathbf{B}\mathbf{X^-1}\) and
\(\mathbf{B}=\mathbf{X^-1}\mathbf{A}\mathbf{X}\).
pg 149 - T15: T15y Suppose that A and B are similar matrices of size n. Prove that \(A^3\) and \(B^3\) are similar matrices. Generalize.
\(\mathbf{A^3}=(\mathbf{X}\mathbf{B}\mathbf{X^-1})(\mathbf{X}\mathbf{B}\mathbf{X^-1})(\mathbf{X}\mathbf{B}\mathbf{X^-1})\)
\((\mathbf{X}\mathbf{B}\mathbf{X^-1})(\mathbf{X}\mathbf{B}\mathbf{X^-1})(\mathbf{X}\mathbf{B}\mathbf{X^-1})=(\mathbf{X}\mathbf{B})(\mathbf{X^-1}\mathbf{X})(\mathbf{B})(\mathbf{X^-1}\mathbf{X})(\mathbf{B}\mathbf{X^-1})\)
\((\mathbf{X}\mathbf{B})(\mathbf{X^-1}\mathbf{X})(\mathbf{B})(\mathbf{X^-1}\mathbf{X})(\mathbf{B}\mathbf{X^-1})=(\mathbf{X}\mathbf{B})(I)(\mathbf{B})(I)(\mathbf{B}\mathbf{X^-1})\)
\((\mathbf{X}\mathbf{B})(I)(\mathbf{B})(I)(\mathbf{B}\mathbf{X^-1})=(\mathbf{X}\mathbf{B^3})(I)(\mathbf{X^-1})\)
So, \(A^3\) and \(B^3\) are similar matrices. I suppose if you were to generalize, you could say that a matrix \(A^n\) is similar to a matrix \(B^n\) so long as n is less than or equal to zero.
pg 149 - T16: Suppose that A and B are similar matrices, with A nonsingular. Prove that B is nonsingular, and that \(\mathbf{A}^-1\) is similar to \(\mathbf{B}^-1\).
If A is nonsigular, then \(\mathbf{A^-1}=(\mathbf{X}\mathbf{B}\mathbf{X^-1})^-1=(\mathbf{X^-1})^-1(\mathbf{B^-1)(\mathbf{X^-1})}=\mathbf{X}\mathbf{B^-1}\mathbf{X^-1}\) which shows that B is nonsingular.