Similar matrices and main diagonal summation?

[zjohnson19]

Homework Statement

True or False? If A is an n × n matrix, P is an n × n invertible matrix, and B = P −1AP, then
a11 + a22 + . . . + ann = b11 + b22 + . . . + bn

Homework Equations

Diagnolization, similar matrixes

The Attempt at a Solution

the question is asking if the summation of the main diagnols of A and B are the same. B is known to be similar to A since B = P^-1 AP. I can't find a counterexample, so I am assuming the summation of the diagnols of both are in fact equal, but this is hardly a proof. I know A and B share the same determinant, rank, and eigenvalues and rank, but I'm not sure how these relate to the main diagnol of the matrices.

 

[blue_leaf77]

Are you familiar with the concept of trace of a matrix?

 

[Twigg]

The key is that A and B have the same characteristic polynomial. The sum of elements on the main diagonal (called the trace of the matrix) is the first-order coefficient of the characteristic polynomial. To prove that you can either use induction or start with the first order Vieta formula and prove that the trace equals the sum of the eigenvalues.

 

[blue_leaf77]

The trace of a matrix $A$ is the sum of the diagonal elements, namely $\textrm{Tr }A = \sum_{i=1}^n A_{ii}$. This means the trace of a product of two matrices will be $\textrm{Tr }AB = \sum_{i=1}^n \sum_{j=1}^n A_{ij}B_{ji}$.

Since you have a product of three matrices, the the trace reads
$$
\textrm{Tr }ABC = \sum_{i=1}^n \sum_{j=1}^n (AB)_{ij}C_{ji}
$$
Now write $(AB)_{ij}$ in terms of the sum of products between the elements of $A$ and $B$.