Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Apart from simplifying matrix powers, why do want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices. Thanks for any answers.

share|cite|improve this question
In addition to answers, see also some suggestions at – Alexander Konovalov Mar 22 '15 at 13:44
up vote 5 down vote accepted

An important motivation for the determination of eigenvectors and eigenvalues of a (square) matrix is that it helps understanding the geometry of the underlying linear transformation.

As an example, consider for instance that the analysis of the eigenvalues of a $3\times3$ orthogonal matrix leads immediately to the classical result that a rigid motion of the $3$-dimensional space that leaves a point fixed is a rotation around some axis.

share|cite|improve this answer

Computing powers of the matrix is just part of it -- what is important that it is simple to understand what the matrix does when you view it from a diagonalizing basis. Each coordinate of the input vector simply gets multiplied by the corresponding diagonal element, and there are no cross-term between different coordinates.

One important application of this is if you have a vector differential equation $X'(t) = AX(t)+B$. Here, if you can switch to a basis that diagonalizes $A$, the equation decouples into independent differential equation for each coordinate, which are easily solvable. (This is even more important because higher-order ODEs in a single variable can be rewritten as a first-order vector equation and solved by the same process).

share|cite|improve this answer
I cannot add anything here because, @Henning note almost all you need. But, I can't imagine what would we have if the identity matrix was lost suddenly. A world without $\text{id}$??? – Babak S. Dec 16 '12 at 16:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.