Matrices are a vital tool that can cross over across many different areas of mathematics. And here we are going to learn the great skill of using them to solve systems of ODEs. Our method will use a few special objects for matrices, Eigenvalues and Eigenvectors. And I will give a quick explanation of these before setting out on the overall problem below!

What are Matrices?

Wikipedia says that “a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.” But personally, I prefer to think of them as a tool for solving lots of small confusing problems in one fell swoop. They take the form of a kind of 2-d vector and are used by almost every area of higher maths. My teacher once told me, “The sooner you learn to love matrices, the sooner you learn to love maths” and I believe this encapsulates it perfectly.

Eigenvalues and Eigenvectors

The words Eigenvalue and Eigenvector are derived from the German word eigen, meaning special, specific or characteristic, and this defines pretty much what they each are. They are special numbers or vectors that relate to a specific matrix. They are intrinsic values that can be calculated and used to properly understand a matrix. Eigenvectors remain at the same angle when transformed under the matrix, and the eigenvalues define the scale factor for these.

To calculate the eigenvalue, we can use a nice rule of setting the determinant of our matrix minus the eigenvalue on the diagonal, then setting this equal to zero. For a two by two matrix this will yield a quadratic equation that can be factorised asbelow or you can use the quadratic formula. It is common to use the symbol lambda (λ) to represent the eigenvalue.

finding Eigenvalues of matrix (made with Latex)

For eigenvectors we can just use its definition to calculate it. Note that there will be and infinite family of them for each eigenvalue of the matrix.

finding Eigenvectors of matrix (made with Latex)

These we can use to find two eigenvectors. Personally, I like to keep A as one and find a B to work for that, giving:

Into our Solution Development

First, we can transform our system into a matrix equation.

turning System to matrix ODE (made with Latex)

Now, we can identify the solution, since it is just a simple exponential differential equation, with solution:

solving the matrix ODE (made with Latex)

If this step does not come naturally, then I would suggest substituting variables for each of the objects and trying to solve it then, remembering that our parameter is ‘t’.

From here all we need to do is compute this matrix exponent, which we can do using a trick called diagonalisation along with our eigenvalues and eigenvectors from before.

Diagonalisation

For this we must convert our single matrix into three separate ones multiplied together. The first is filled with the eigenvectors from before and I will name it ‘T’:

The second is filled with our eigenvalues on the diagonal with 0s everywhere else and I will name it ‘D’

The final one will be the inverse of T, with we can compute using the two by two matrix inverse rule, giving:

Inverse Transformation Matrix (made with Latex)

Using these three matrices, we can compute any “analytic” function of it, such as the exponential function or powers. It just so happens that when we now perform a function on these three matrices multiplied together, we get:

Diagonalisation Rule for Matrices (made with Latex)

And since with diagonal matricies, any function can just be brought onto the individual terms, we can now solve our problem!

From here, all we must do is multiply our three matrices together and receive once result:

multiplying out the matrices (made with Latex)

Using this, we can get our two equations for y and x as:

Final Thoughts

As you can see through this demonstration, matrices are a fundamental area of maths that can be used even when it looks at first glance like they can’t. They teach use to look around the box not into it and examine all the exquisite beauty of the problem before we tackle it. If you found this informative, I would suggest you, switch up the numbers in our original equation and try it out for yourself. And as always,

Have fun and never stop puzzling.

Using Matrices to solve a System of ODEs was originally published in Cantor’s Paradise on Medium, where people are continuing the conversation by highlighting and responding to this story.