# Elementary Linear Algebra - Similar Matrices and Rank

## Homework Statement

Suppose matrices A and B are similar. Explain why they have the same rank.

## The Attempt at a Solution

So if A and B are similar, then there is some invertible matrix P such that B = P^-1AP. I have been trying to find some way to relate rank(A) to rank(P^-1AP) but I can't figure it out. I feel like maybe i'm missing some intuition about this. This comes at the end of a section about linear transformations.

Thank you to anyone who can help.

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
What is the definition of "rank"?

"The dimension of the row (or column) space of a matrix is called the rank of A and is denoted by rank(A)."

Thinking about this I recall that the rank of a linear transformation is equal to the rank of the matrix for T (where T is defined by T(x) = Ax). So if A and B both represent the same linear transformation but with respect to different bases, then they will have the same rank. This relationship is described by them being similar. P and P^-1 are the transition matrices between the bases.

Is my thinking headed in the right direction? I still feel unclear about this. Is there a way to discuss this without reference to transformations and just from the definition of being similar?

Thank you for your time.

The rank of A is the dimension of the image of A. So take a basis of the image of A. Can you maje this into a basis of the image of $PAP^{-1}$??