Unique Positive Definite Square Root of a Positive Definite Matrix

Question

If $A$ be an $n\times n$ positive definite matrix, then there exists a unique positive definite matrix $B$ such that $B^2=A$.

My question is how to get this $B$. What is the name of the algorithm for finding $B$?

Answer 1

The positive definite matrix $A$ is diagonalizable. Thus, it can be written in the form $A = V \cdot D\cdot V^{-1}$, where $D$ is a diagonal matrix with the positive eigenvalues on the main diagonal and $V$ contains $n$ linearly independent right hand eigenvectors as its columns. Actually, we can consider the matrix $V$ to be an orthogonal one (i.e. $V^{-1} = V^T$).

So: $$B=A^{1/2} = V\cdot D^{1/2}\cdot V^{-1} = V\cdot D^{1/2}\cdot V^T$$

In that case, $B$ is a positive definite matrix. Indeed, consider any vector $x\neq \mathbf 0$. Then: $$x^TBx=x^T VD^{1/2}V^Tx = (x^T V) \cdot D^{1/2} \cdot (x^TV)^T >0, $$ due to $D^{1/2}$ is a positive definite matrix.

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Yes, this method uses diagonalization of matrix $A$.