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[parent] square root of positive definite matrix (Definition)

Suppose $ M$ is a positive definite Hermitian matrix. Then $ M$ has a diagonalization

$\displaystyle M= P^* \operatorname{diag}(\lambda_1, \ldots, \lambda_n) P $
where $ P$ is a unitary matrix and $ \lambda_1, \ldots, \lambda_n$ are the eigenvalues of $ M$, which are all positive.

We can now define the square root of $ M$ as the matrix

$\displaystyle M^{1/2}= P^* \operatorname{diag}(\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n}) P. $
The following properties are clear
  1. $ M^{1/2} M^{1/2}=M$,
  2. $ M^{1/2}$ is Hermitian and positive definite.
  3. $ M^{1/2}$ and $ M$ commute
  4. $ (M^{1/2})^T=(M^T)^{1/2}$.
  5. $ (M^{1/2})^{-1}=(M^{-1})^{1/2}$, so one can write $ M^{-1/2}$
  6. If the eigenvalues of $ M$ are $ (\lambda_1, \ldots, \lambda_n)$, then the eigenvalues of $ M^{1/2}$ are $ (\sqrt{\lambda_1}, \ldots, \sqrt{\lambda_n})$.



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"square root of positive definite matrix" is owned by rspuzio. [ full author list (4) | owner history (1) ]
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See Also: Cholesky decomposition


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Cross-references: clear, properties, matrix, positive, eigenvalues, unitary matrix, diagonalization, Hermitian matrix, positive definite
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This is version 9 of square root of positive definite matrix, born on 2005-05-17, modified 2007-10-12.
Object id is 7067, canonical name is SquareRootOfPositiveDefiniteMatrix.
Accessed 7934 times total.

Classification:
AMS MSC: 15A48 (Linear and multilinear algebra; matrix theory :: Positive matrices and their generalizations; cones of matrices)
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