Non-Linear Transformations

Definition

1. Matrix Exponentiation:

   • Raising a matrix to a power greater than one, as in �2A2 or ��An, is not a linear transformation. The composition of linear transformations is not equivalent to matrix exponentiation.

2. Matrix Determinant:

   • Taking the determinant of a matrix, denoted as $\text{det}\left(A\right)$, is not a linear transformation. The determinant does not satisfy the additivity property.

3. Matrix Inversion (in general):

   • While matrix inversion itself is considered a linear transformation ((�−1)�=�−1�(A−1)v=A−1v), it is not applicable to all matrices. For non-invertible matrices, the inverse does not exist, and the operation is undefined.

4. Matrix Logarithm:

   • Taking the logarithm of a matrix, denoted as $\log \left(A\right)$, is not a linear transformation. The composition of linear transformations is not equivalent to matrix logarithm.

5. Element-Wise Operations:

   • Operations that apply nonlinear functions element-wise to the matrix entries, such as taking the square root or applying trigonometric functions to individual elements, are not linear transformations.