More Than Just the Rotation Matrix

Matrices, an important function in both calculus and linear algebra, can be defined geometrically as a method of writing down the vectors of a coordinate system. So, by definition, a 2x2 matrix is the graph of a parallelogram, formed by the vectors of each column, drawn from the origin of a cartesian plane.

For example, say we have matrix A such that:

Then, we know that A can be drawn as a parallelogram with vectors:

and

drawn from the origin.

Matrices have countless uses; including quantum mechanics, engineering, and programming, and we can manipulate its elements so that the matrix can achieve a specific goal.

One of the most famous matrices is the Identity, or ‘unit’ matrix. Represented by the letter ‘I’, this function is similar to the number 1 in integer multiplication.

If you multiply any vector or matrix by the identity matrix, you get the value of the multiplicand.

But the identity matrix is not the only special matrix - we can create matrices for transformations, translations, reflections, or even rotations. If you multiply a point, vector, or another matrix by the rotation matrix,

the result will be a θ° counterclockwise rotation of your original image.

So, if you wanted a matrix A such that it rotates every image by 90° counter clockwise, you would plug in 90° for θ to get:

since cos(90) = 0 and sin(90) = 1.

While this alone is useful, what if there is something else that the rotation matrix can be used for?

If we take a matrix that rotates by (a+b) counterclockwise,

we know that we can break the rotation into two different matrices - a counterclockwise rotation by (a) and a counterclockwise rotation by (b), resulting in the following product:

Thus, we have proved the trigonometric identities:

cos(a+b) = cos a cos b - sin a sin b

and

sin(a+b) = sin a cos b+ cos a sin b

with matrices.

These are but a few of the uses for matrices, but there are so many more!

What is your favorite matrix/use of matrices?