📂Matrix Algebra

Square Root Matrix

Definition ¹

A matrix $A$ is called the Square Root Matrix of $B$ if it satisfies the following condition and is denoted by $\sqrt{A} := B$. $$B^{2} = A$$

Description

The concept of square roots becomes more interesting in the context of matrices. For instance, $$A = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$$ if there is a matrix like this, its square root matrix is not $$\begin{bmatrix} \sqrt{2} & \sqrt{2} \\ \sqrt{2} & \sqrt{2} \end{bmatrix}$$ where each element is the square root of the given element, but rather it is a matrix where all elements are $1$ like this: $$\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$$

Imaginary Matrix

Although it is not very important in the real world of mathematics, there has been a competition studying Imaginary Matrix, which is the square root matrix of $-E$, i.e., $$X^{2} = - E_{p}$$ satisfying $X$, held at a shrimp sushi restaurant.