Eigenvalues and Eigenvectors are used in many engineering and science applications such as Control theory, vibration analysis, electric circuits etc. Many applications use eigenvalues and vectors to transform a given matrix into a diagonal matrix.

Matrices with distinct eigenvalues have eigenvectors which are linearly independent (L.I). A matrix P formed by these L.I eigenvectors as its column vectors thus will have its determinant, det(P) != 0. So, that means the inverse of matrix P exists. This P matrix is called the modal matrix of A. Moreover, the product P`AP is a diagonal matrix, D. (P` = inverse of P).

This diagonal matrix D has the eigenvalues of A as its diagonal elements. Therefore, D has the same eigenvalues as matrix A and thus matrix A ad D are said to be similar.

Since D = P`AP, where P` = inverse of P, we can obtain A,

A = PDP`

Obtaining high powers of matrix A involves many multiplications. However, using the above formula for A, we can obtain powers of A easily.

A^2 = A.A = (PDP`)(PDP`) = PD.D.P` = P(D^2)P`

and similarly,

A^k = P(D^k)P`

Now we can show how this process is important to solve couple differential equations of both first and second order.

Case 1)

If we have the following differential equations,

x’ = -2x

y’ = -5y where (a’ is the derivative of a)

then, this can be represented by the matrix notation of

|x’| = |-2 0 | |x|

|y’|    | 0 -5| |y|

This is of the form X’ = AX

Case 2)

If we have the following differential equations,

x’ = 4x + 2y

y’ = -x + y

this can be represented by the matrix notation of

|x’| = |4  2||x|

|y’|    |-1 1||y|

which is also in the form X’ = AX

Now you can see that equation pair of case 1 is separate which means that equation 1 only involved unknown x and equation 2 only involved unknown y. Thus, this corresponds to the diagonal matrix A we obtained in case 1.

Case 2 on the other hand has coupled equations because both equations involve x and y. Thus we get a non-diagonal matrix for A.

It is clear that the second case is more difficult to solve and thus we use our knowledge of digitalization.

So now consider a system of differential equations in the matrix form of

X’ = AX where

X’= |x’(t)| and X = |x(t)|

       |y’(t)|                |y(t)|

We introduce a new column vector of unknowns

Y = |r(t)|

       |s(t)|

through the relation,

X = PY, where P is the modal matrix of A. So,

X’ = PY’ (because P is a matrix of constants).

But, X’ = AX. Hence, PY’= AX and thus, PY’ = A(PY) (because X = PY)

Now by multiplying both sides by P` (which is the inverse of matrix P) we get,

Y’ = (P`AP)Y

Because of the properties of the modal matrix, P`AP is a diagonal matrix D, which again means that the distinct eigenvalues of matrix A will be the diagonal entries of D.

Thus, if we have µ1 and µ2 as the distinct eigenvalues of A then,

P`AP = | µ1  0 |

              | 0  µ2 |

Hence,

Y’ = (P`AP)Y becomes,

|r’| = |µ1  0| |r|

|s’|     |0  µ2| |s|

Now we have,

r’ = µ1r

s’ = µ2s

Notice that these equations are decoupled!

We can easily obtain the solutions for r and s from the above equations.

r(t) = C1e^ µ1t

s(t) = C2e^ µ2t where C1, C2 are arbitrary constants.

Now, we can use the values of r(t) and s(t) for our relation X = PY. Thus, we can now easily find the unknowns x and y.

The only restriction for this method is that matrix A in the system, X’ = AX should have distinct eigenvalues.

[Sources]

Linear Alebra with Applications - Otto Bretscher

http://www.cems.uwe.ac.uk/~kgolden/HELM/HELM%20Sept%2005%20(D)/pages/workbooks_1_50_sep2005/workbook_22/22_2_applctn_eignval_eignvec.pdf