Matrix Squares

Definition

The square of a square matrix is the product of the matrix with itself. A square root (or matrix square root) of a given matrix is a matrix whose square is the given matrix. A matrix is idempotent if it is equal to its square.

Notation

Let $A \in \R ^{n \times n}$. Then the square of $A$ is $AA$. We denote the square of $A$ by $A^2$. $A$ is idempotent if $A^2 = A$. $B \in \R ^{n \times n}$ is a square root of $A$ if $A = B^2$.

Existence and uniqueness

Clearly a matrix can have a square root. Take for example the matrix in $\R ^{1 \times 1}$ which is $\bmat{1}$. A square root of this matrix is $\bmat{1}$, but also $\bmat{-1}$. So matrix square roots do exist, but are not unique.