Matrix Differential Equation - Solved Example of A Matrix ODE - First Step

First Step

The first step, that has already been mentioned above, is finding the eigenvalues. The process of finding the eigenvalues is not a very difficult process. Both eigenvalues and eigenvectors are useful in numerous branches of mathematics, including higher engineering mathematics/calculations(i.e. Applied Mathematics), mechanics, physical mathematics, mathematical economics, and linear algebra.

Therefore, the process consists of the following:

=

The derivative notation y' etc. seen in one of the vectors above is known as Lagrange's notation, first introduced by Joseph Louis Lagrange. It is equivalent to the derivative notation dy/dx used in the previous equation, known as Leibniz's notation, honouring the name of Gottfried Leibniz.

Once the coefficients of the two variables have been written in the matrix form shown above, we may start the process of evaluating the eigenvalues. To do that we are going to have to find the determinant of the matrix that is formed when an identity matrix, multiplied by some constant lambda, symbol λ, is subtracted from our coefficient matrix in the following way:

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Applying further simplification and basic rules of matrix addition we come up with the following:

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Applying the rules of finding the determinant of a single 2×2 matrix, we obtain the following elementary quadratic equation:

which may be reduced further to get a simpler version of the above:

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Now finding the two roots, and of the given quadratic equation by applying the factorization method we get the following:

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The values, and, that we have calculated above are the required eigenvalues. Once we find these two values, we proceed to the second step of the solution. We'll use the calculated eigenvalues later in the final solution. In some cases, say other matrix ODE's, the eigenvalues may be complex, in which case the following step of the solving process, as well as the final form and the solution, dramatically change.