Applying (pre multiplying) a matrix to a vector gives you either a rotation or a scale(strech/compress) or both a scale(strech/compress) and a rotation.
This is called vector transformation, and the (pre multiplying) matrix is called the transformation matrix.
No Rotation Case(Pure Scaling ; EigenValue/Eigen Vector)
When a vector transformation by a matrix is just the right match as if it was scaled without any roation the scaler is called the eigenvalue and the vector is called the eigenvector.
The equation is: Av=λV(matrix A pre multiplying vector v is the same as lambda( scalar ) multiplying vector v)
No Scaler Case(Pure Rotation)
When multiplying a vector with the following matrix(A.K.A the rotation matrix) with an angle θ of any value the vector will rotate counter clockwise.
We know that the magnitude of the vector that gets applied to each element is equal to 1 and thus there is no scaling. We know this by calculating the magnitude using the Euclidean formula from the pythagoras rules of cosines/sines we know when √(cos² + sin² )=√(1) and cancels out any scaling.
