Diagonalisation of Matrices

matqkks

What is the purpose of diagonalisation of matrices?
Why do teach this stuff?
Is there any serious tangible application of diagonalisation?
Do engineers or physics need this process?

Klockan3

1.To better understand what the linear operator represents, the act of diagonalization is the act of putting the matrix into its natural basis where its properties become trivial.
2.Because it is really important.
3.Yes, just about all modern science is built upon it since it is crucial for differential equations among other things.
4.Yes.

mathwonk

diagonalization is just one example of finding a simple matrix for a given linear map. A linear map has one matrix representation for each choice of basis. The geometry of that map is the same for all bases and hence all matrices, but that geometric behavior is harder to see in some bases than in others.

thus we want to choose among all bases the one that reveals as clearly as possible the geometric behavior of the map. For those maps that have a diagonal representation, that diagonal representation is the simplest possible and not only allows a clear picture of the geometric behavior of the map but also allows easy algebraic computation of powers and even polynomials in the map.


so to put it another way, asking why we want to know how to diagonalize a matrix is similar to asking why we want to be able to visualize the action of the map. I.e. if we want to understand the map we want to know how to diagonalize it, when possible. It also helps in differential equations, as observed above, since there we want to solve he equations, and since we know how to solve diagonal equations, diagonalizing equations allows us to solve them.


by analogy, completing the square allows us to solve quadratic equations by reducing them to a simpler form. that's all that is going on here, we are finding a simpler representation of the matrix that makes it easier to understand it, compute with it, and to solve equations using it.