# Diagonalization of linear transformations

Hello everybody!!
My questions are about diagonalization of linear transformations:
1. What are the motivations for diagonalize linear transformations?
2. How does diagonalizing a linear transformation help in theoretical mathematics and applied mathematics?

For example, my way of interpreting or motivating the idea of eigenvalues and eigenvectors is as follows: Linear transformations[$T$ ] (for example, in $\mathbb{R}^{3}$ to fix ideas) are those applications that "transform" certain vector spaces into others; but it is not just any transformation, but those that preserve the algebraic structure of said vector spaces. Now, in this action of "transforming" it happens that the vectors, elements of the vector spaces, change direction; but other vectors maintain the same direction, but have been lengthened or shortened or worse still changed direction (but still maintain their direction). For example: let be a nonzero vector $v\in \mathbb{R} ^{3}$ be a vector before applying the linear transformation $T$ and $u\in \mathbb{R} ^{3}$ the resulting vector after the transformation; in other words, $T(v)=u=\lambda v, \lambda \in \mathbb{R}$
Now, these types of vectors that do not change direction are of great importance in both theoretical and applied mathematics.
But, I don't have a motivation to study the diagonalization of linear transformations.

Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.