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Section 1.7 Normal Matrices

Normal matrices comprise a broad class of interesting matrices, many of which you probably already know by other names. But they are most interesting since they define exactly which matrices we can diagonalize via a unitary matrix. This is the upcoming Theorem [provisional cross-reference: orthonormal diagonalization].

Definition 1.7.1. Normal Matrix.

The square matrix \(A\) is normal if \(\adjoint{A}A=A\adjoint{A}\text{.}\)

So a normal matrix commutes with its adjoint. Part of the beauty of this definition is that it includes many other types of matrices. A diagonal matrix will commute with its adjoint, since the adjoint is again diagonal and the entries are just conjugates of the entries of the original diagonal matrix. A Hermitian (self-adjoint) matrix (Definition HM) will trivially commute with its adjoint, since the two matrices are the same. A real, symmetric matrix is Hermitian, so these matrices are also normal. A unitary matrix (Definition UM) has its adjoint as its inverse, and inverses commute (Theorem OSIS), so unitary matrices are normal. Another class of normal matrices is the skew-symmetric matrices. However, these broad classes of matrices do not capture all of the normal matrices, as the next example shows.

Consider the matrix \(\begin{bmatrix}1 & -1\\1 & 1\end{bmatrix}\text{.}\) Then

\begin{equation*} \begin{bmatrix}1 & -1\\1 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\-1 & 1\end{bmatrix} = \begin{bmatrix}2 & 0\\0 & 2\end{bmatrix}=\begin{bmatrix}1 & 1\\-1 & 1\end{bmatrix}\begin{bmatrix}1 & -1\\1 & 1\end{bmatrix} \end{equation*}

so we see by Definition Definition 1.7.1 that \(A\) is normal. However, notice that \(A\) is not symmetric (hence, as a real matrix, not Hermitian), not unitary, and not skew-symmetric.