Is every diagonal matrix the product of 3 matrices, $P^{-1}AP$, and why? In trying to figure out which matrices are diagonalizable, why does my textbook pursue the topic of similar matrices?
It says that "an $n \times n$ matrix A is diagonalizable when $A$ is similar to a diagonal matrix. That is, $A$ is diagonalizable when there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix." I do not understand why it would begin with considering similar matrices. I mean, what is the motivation?
 A: We say that two square matrices $A,B$ are similar if there exists an invertible matrix $P$ such that $A = P^{-1} BP$.
More to the point, however, two matrices are similar if they represent the same linear transformation viewed in different coordinate systems.
Let us write the above equation as
$$ PA = BP $$
Therefore $PAx = BPx$. If $P$ is my change of coordinates matrix, then this equation says that the following two procedures give the same result:


*

*Multiply $x$ by $A$, then change coordinates using $P$.

*Change the coordinates of $x$ using $P$, then multiply the result by $B$.


So diagonalizability is a statement about what a matrix $A$ "looks like" in a certain coordinate system. When you multiply a vector $x$ by a diagonal matrix $D$, it just multiplies the components of $x$ by the corresponding diagonal entries. So if $D = P^{-1}AP$, then this says that in the coordinates given by the matrix $P$, multiplication by $A$ just scales each component of a vector (written in the $P$-basis) by some fixed value.
A: Matrices are connected to linear maps of vector spaces, and theres a concept of a basis for vector spaces. (A basis is something so that every element is a can be written uniquely as a sum of the elements in the basis.)
Now, bases aren't unique, so if you want to see what your matrix looks like under  a different basis, this is equivalent to conjugating by the change of basis matrix as you've written above.
Now a matrix is said to be diagonalisable if there is some basis under which the matrix becomes diagonal.
If you read further on in your textbook, it should tell you all this.
