Motivation behind concept of diagonalization of a linear operator What is the motivation behind the concept of diagonalization of linear operator? Why mathematicians introduced this concept? 
 A: If a linear operator can be diagonalized then problems involving it can be simplified a great deal.  Specifically that means that we can "uncouple" the equations.  If we have, say, 50 equations in the 50 unknowns, $x_1$, $x_2$, $\cdot\cdot\cdot$, $x_{50}$. "Diagonalizing" the matrix of coefficients "uncouples" the equations so that we still have 50 equation but each equation involves only one of the unknowns, which can each be, very simply, solved.
A: Geometrically, when an operator can be diagonalized, this says that the operator is just scaling along a certain set of directions, by an amount depending on the direction. We can see this since $$A=SDS^{-1}\Leftrightarrow AS=SD \Leftrightarrow As_{i}=d_{i}s_{i},\;1\leq i\leq n,$$ where $s_{i}$ is the $i$th column of $S$ and $d_{i}$ is the $i$th diagonal entry of the diagonal matrix $D$.
Then we may interpret the operation of $A$ on any vector as just the composition of these scalings. Given $x,$ $x=S(S^{-1}x)=\sum_{i=1}^{n}(S^{-1}x)_{i}s_{i}=\sum_{i=1}^{n}c_{i}s_{i},$ which is a linear combination of the vectors $s_{i}.$ Then $$Ax=\sum_{i=1}^{n}c_{i}(As_{i})=\sum_{i=1}^{n}c_{i}d_{i}s_{i},$$ which just performs these scalings on the various components.
A: Because a diagonal matrix is easy to manipulate and study, mathematicians started wondering over which conditions a matrix or linear operator can be represented by a diagonal matrix. This way lead them to build all the theorems,theory and algorithms regarding the conditions in order to represent a linear operator or a square matrix as a diagonal matrix.
