Another way of looking at it is that diagonalizing a matrix "uncouples" the equations.
A general matrix can be thought of a representing a system of linear equations. If that matrix can be diagonalized, then we have the same number of equations but each equation now has only one of the unknown values in it. For example, the matrix equation
Ax= b= \begin{bmatrix}-1 & 6 \\ -4 & 6\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}2 \\ 3 \end{bmatrix}.
That matrix, A, has eigenvalues 2 and 3 with corresponding eigenvectors <2, 1> and <3, 2> respectively. Let P= \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}. Then P^{-1}= \begin{bmatrix}2 & -3 \\ -1 & 2\end{bmatrix} and P^{-1}AP= \begin{bmatrix}2 & -3 \\ -1 & 2\end{bmatrix}\begin{bmatrix}-1 & 6 \\ -4 & 6\end{bmatrix}\begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}= \begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}
Now we can rewrite the equation Ax= b as A(PP^{-1})x= b so (P^{-1}AP)P^{-1}x= P^{-1}b If we let z= P^{-1}x[/tex], here, z= \begin{bmatrix}z_1 \\ z_2\end{bmatrix}= \begin{bmatrix}2 &amp; -3 \\ -1 &amp; 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}, then P^{-1}b= \begin{bmatrix}2 &amp; -3 \\ -1 &amp; 2\end{bmatrix}\begin{bmatrix}2 \\ 3\end{bmatrix}= \begin{bmatrix}-5 \\ 4\end{bmatrix} so the matrix equation becomes \begin{bmatrix}2 &amp; 0 \\ 0 &amp; 3\end{bmatrix}\begin{bmatrix}z_1 \\ z_2\end{bmatrix}= \begin{bmatrix}-5 \\ 4\end{bmatrix} which is exactly the same as the two equations 2z_1= -5 and 3z_1= 4 which are "uncoupled"- they can be solved separately. After you have found z, because z= P^{-1}y, y= Pz= \begin{bmatrix}2 &amp; 3 \\ 1 &amp; 2\end{bmatrix}\begin{bmatrix}z_1 \\ z_2\end{bmatrix}.<br />
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Of course the work involved in finding the eigenvalues and eigenvectors of a matrix and diagonalizing, here, is far more than just solving the equations- but think about a situation where your system has, say, 1000 equations in 1000 unknown values. Also, it is not uncommon that applications involve solving many systems of the form Ax= b, each with the <b>same</b> A and different bs. In a situation like that,the diagonalization only has to be done <b>once</b> for the whole problem.<br />
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Also, there are important situations where we have systems of linear <b>differential equations</b>. The same things apply there- diagonalizing "uncouples" the equations.<br />
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As I said before, not every matrix <b>can</b> be "diagonalized"- but every matrix can be put in "Jordan Normal Form", a slight variation on "diagonalized" where we allow some "1"s just above the main diagonal, which <b>almost</b> uncouples the equations- no equation involves more then two of the unknowns.