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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.

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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.

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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).

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    $\begingroup$ 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 $\text{id}$??? $\endgroup$ – mrs Dec 16 '12 at 16:32

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