Square root of a positive semidefinite matrix

Definition

Let $A$ be a positive semidefinite $n \times n$ complex Hermitian matrix. There is exactly one positive semidefinite matrix $B$ with $B^2 = A$. This matrix is the square root of $A$, written $A^{1/2}$ or $\sqrt{A}$.

Remarks

If $A = U D U^*$ with $U$ unitary and $D$ diagonal with entries $d_i \ge 0$, then $A^{1/2} = U D^{1/2} U^*$, with entries $\sqrt{d_i}$ in $D^{1/2}$. $A^{1/2}$ commutes with each matrix that commutes with $A$. Without the condition $B \succeq 0$, a matrix can have many square roots; for example, $I_2$ has infinitely many.

Explanations

L1 human

A positive semidefinite matrix has exactly one positive semidefinite square root. In an orthonormal basis of eigenvectors, we take the square root of each eigenvalue. This is like the positive square root of a nonnegative number.