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Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix–a so-called diagonal matrix–that shares the same fundamental properties of the underlying matrix. Matrix diagonalization is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes this canonical form. Diagonalizing a matrix is also equivalent to finding the matrix’s eigenvalues, which turn out to be precisely the entries of the diagonalized matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix.

The remarkable relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical identity (the eigen decomposition) that a square matrix A can be decomposed into the very special form

 A=PDP^(-1),
(1)

where P is a matrix composed of the eigenvectors of A, D is the diagonal matrix constructed from the corresponding eigenvalues, and P^(-1) is the matrix inverse of P. According to the eigen decomposition theorem, an initial matrix equation

 AX=Y
(2)

can always be written

 PDP^(-1)X=Y
(3)

(at least as long as P is a square matrix), and premultiplying both sides by P^(-1) gives

 DP^(-1)X=P^(-1)Y.
(4)

Since the same linear transformation P^(-1) is being applied to both X and Y, solving the original system is equivalent to solving the transformed system

 DX^'=Y^',
(5)

where X^'=P^(-1)X and Y^'=P^(-1)Y. This provides a way to canonicalize a system into the simplest possible form, reduce the number of parameters from n×n for an arbitrary matrix to n for a diagonal matrix, and obtain the characteristic properties of the initial matrix. This approach arises frequently in physics and engineering, where the technique is oft used and extremely powerful.

Reference : MathWorld

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