Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

Question

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors.

However, the importance of diagonalizing a linear map or a bilinear form and the significance (and physical meaning) of eigenvalues and eigenvectors has never been properly explained to me.

Could you clarify these points (also by pointing out some references)?

Answer 1

As a general reference may I suggest Wikipedia... also I am not going to address bilinear forms.

For the question about eigenvalues and eigenvectors please see What is the importance of eigenvalues/eigenvectors?

If a linear map on an $n$-dimensional vector space $T:V\rightarrow V$ is diagonalised then it may be represented as a diagonal matrix. A diagonal matrix is of the form

$$T=\left(\begin{array}{cccc}\lambda_1 & 0 & \cdots & 0 \\0 & \lambda_2 & \cdots &0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{array}\right).$$

If you know how a matrix represents a linear map then you will know that this implies that we have written $T$ with respect to an ordered basis of $V$, $\mathcal{B}=\{e_1,e_2,\dots,e_n\}$, such that each of the elements of $\mathcal{B}$ are eigenvectors of $T$:

$$Te_i=\lambda_i e_i.$$

A little more can be said but this is a good initial foray.

Answer 2

The best physical intuitive and interactive explanation of eigenvalues/eigenvectors can be found in the link below

http://setosa.io/ev/eigenvectors-and-eigenvalues/