Why is a matrix diagonalizable

It may be that for some collection of vectors it does scaling whereas for others it does shifting, or it can also do linear combinations of these actions block scaling and shifting simultaneously. This idea is the main content behind the Jordan normal form. Being diagonalizable means that it does not do any of the shifting, and only does scaling. Otherwise this is not the case. I'll try an answer in a different equivalent direction: what happens when the matrix is not diagonalizable?

First of all, this must mean that some of the matrix's eigenvalues occur more than once. So what if two eigenvalues are equal? But the important point is that it doesn't crush it to zero! Instead it only crushes it to some subspace. In general, if you have a nilpotent matrix all eigenvalues vanish there are many subspaces of varying dimensions to pick from and so many different ways to crush the space.

The precise position of those ones if there are any determines which subspace is being crush to which and so on.

Having said all this, it would be a sin now not to mention Jordan decomposition. But in general it can perform a lot of nontrivial shuffling corresponding to the nilpotent part. This happens as a result of stretching. It really bends a light ray by its refractive index along input and output planes.

On the line of thought this may play the role of light ray invisibility by the Einstein gravity of bending typically applicable in Quantum mechanics. A sort of squeezing or shearing along hypotenuse side.

When all the elements are diagonally shifted the potential becomes zero and not so when more than zero above the diagonalization of directional matrics elements. This may also be called a twisting along axial planes as a function of twisting angle of ray transfer matrics pave the way for magnification as divergence ,convergence as well as for invisible clocking dynamics using laser beams. The behavior of linear dynamical systems, both continuous and discrete, can be expressed in terms of the eigenvalues of the relevant matrix, and the expression and especially the long-term behavior has some added complications if the matrix is not diagonalizable.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Define C as above, so C is invertible by the invertible matrix theorem in Section 5. We multiply by the standard coordinate vectors to find the columns of D :.

Therefore, the columns of D are multiples of the standard coordinate vectors:. Since C is invertible, its columns are linearly independent. We saw in the above example that changing the order of the eigenvalues and eigenvectors produces a different diagonalization of the same matrix.

There are generally many different ways to diagonalize a matrix, corresponding to different orderings of the eigenvalues of that matrix. The important thing is that the eigenvalues and eigenvectors have to be listed in the same order. There are other ways of finding different diagonalizations of the same matrix.

For instance, you can scale one of the eigenvectors by a constant c :. Since A is not invertible, zero is an eigenvalue by the invertible matrix theorem , so one of the diagonal entries of D is necessarily zero. Also see this example below. To diagonalize A :. We will justify the linear independence assertion in part 4 in the proof of this theorem below. Of the following matrices, the first is diagonalizable and invertible, the second is diagonalizable but not invertible, the third is invertible but not diagonalizable, and the fourth is neither invertible nor diagonalizable, as the reader can verify:.

As in the above example , one can check that the matrix. Therefore, up to similarity, these are the only such examples. To prove this, let B be such a matrix. We can compute the first column of A as follows:. Therefore, A has the form. Now we observe that. A diagonal matrix is easy to understand geometrically, as it just scales the coordinate axes:.

Therefore, we know from Section 5. In the following examples, we visualize the action of a diagonalizable matrix A in terms of its dynamics. In other words, we start with a collection of vectors drawn as points , and we see where they move when we multiply them by A repeatedly.

In this subsection, we give a variant of the diagonalization theorem that provides another criterion for diagonalizability. It is stated in the language of multiplicities of eigenvalues. For instance, in the polynomial. We saw in the above examples that the algebraic and geometric multiplicities need not coincide.

However, they do satisfy the following fundamental inequality, the proof of which is beyond the scope of this text. But you'll probably want the answer of your matrix multiplication written w. Example of what I mean:. If anything this seems to be a lot more work. I think, in short, the purpose is more to provide a characterization of the matrix you are interested in, in most cases. A "simple" form such as diagonal allows you to instantly determine rank, eigenvalues, invertibility, is it a projection, etc.

That is, all properties which are invariant under the similarity transform, are much easier to assess. A practical example: principal components is an orthogonal diagonalization which give you important information regarding the independent components eigenvectors in a system and how important each component is eigenvalues - so it allows you to characterize the system in a way which is not possible in the original data.

I can't think of a case where diagonalization is used purely as a means to "simplify" calculation as it is computationally expensive - it is more of an end goal in itself. I'll add that while you mention computing integer powers of matrices, diagonalization helps in computing fractional powers and exponentiation. This significantly reduces the complexity for matrix exponentiation given a required precision.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What's so useful about diagonalizing a matrix? Ask Question.