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# Normal matrices and diagonalizability

Theorem: The product of two unitary matrices is unitary.

Proof: Let and be unitary, i.e., and , then is unitary:

Theorem: Two square matrices and are simultaneously diagonalizable if and only if they commute.

Proof (reference)

• Let and be simultageously diagonalizable by

then

• Let and commute, i.e., . Assuming is an eigenvector of corresponding to eigenvalue , i.e., , then

we see that is also an eigenvector of corresponding to the same eigenvalue , i.e., must be a scaled version of (in the same 1-D space): , i.e., is also an eigenvector of .

Theorem: A matrix is normal if and only if it is unitarily diagonalizable.

Proof (reference)

• If is unitarily diagonalizable:

where is unitary and is a diagonal matrix satisfying , then is normal:

• If is normal, then it is diagonalizable by a unitary matrix. First we show any matrix can be written as

where

are both Hermitian, and diagonalizable by a unitary matrix. As is normal, we have

We see that , i.e., and commute, and they can be simultaneously diagonalized by some unitary matrix :

and so can :

Next: Singular value decomposition Up: algebra Previous: Generalized eigenvalue problem
Ruye Wang 2015-04-27