If is real symmetric and positive semidefinite, its principal square root isIt is symmetric and positive semidefinite, and is positive definite exactly when is.
Every invertible real matrix has the polar decompositionwhere is symmetric positive definite and is orthogonal.
Every invertible complex matrix has the unique polar decompositionwhere is a positive-definite Hermitian matrix and is a unitary matrix. Writing gives , which is Hermitian.
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The square root of a matrix \( A \) is another matrix \( B \) such that when multiplied by itself, it yields \( A \). Mathematically, this is expressed as: \[ B^2 = A \] Not all matrices have square roots, and if they do exist, they may not be unique. The existence of a square root depends on several properties of the matrix, such as its eigenvalues. ### Types of Square Roots 1.