Anyone know matrix decomposition that decompose matrix in to diagonal matrix and hermitian matrix ?

08 August 2015 1 2K Report

Matrix Decomposition, Diagonal and Hermitian

Deffinition. A square matrix A is said to be unitarily diagonalizable if there exists a unitary matrix U such that U∗AU is diagonal matrix.

Normal matrices are exactly those matrices whose Schur decomposition has a diagonal matrix in the middle.

Theorem. (Spectral Decomposition). Let A ∈Mn with eigenvalues λ1,λ2,…λn. Then A is normal (AA∗ = A∗A i.e., real symmetric, Hermitian) if and only if A is unitarily diagonalizable, that is, there exists a unitary matrix U such that

U∗AU = diag(λ1,λ2,…λn).

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