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A complex square matrix $U$ is a unitary matrix when $U^{†}U=I$, equivalently $U^{-1}=U^{†}$. Here $†$ denotes the adjoint matrix, the conjugate transpose. Such a matrix preserves the standard complex inner product and its Euclidean norm.

A real symmetric matrix $A$ is a Hermitian matrix, so $A^{†}=A$. For arbitrary complex $\mathbf{x}$,The imaginary cross terms cancel because $A$ is Hermitian. Reversing the signs gives the same identity:In particular, $(A+iI)\mathbf{x}=0$ implies $|\mathbf{x}{|}^2=0$, so $A+iI$ has zero kernel of a linear map. It is a square matrix, and the rank-nullity theorem therefore makes it invertible. The same argument proves invertibility of $A-iI$.

The proposed matrixis consequently well-defined. For any $\mathbf{y}$, put $\mathbf{x}=(A+iI{)}^{-1}\mathbf{y}$. The two norm identities show directly thatTo verify the unitary matrix identity explicitly, take adjoints:The two shifts commute, because both are polynomials in $A$. HenceThusThe imaginary shifts are invertible, and their equal norm changes cancel in the quotient. This is the Cayley transform of a Hermitian matrix in the $(A-iI)(A+iI{)}^{-1}$ convention.

A complex square matrix $A$ is a normal matrix if it commutes with its conjugate transpose:By unitary diagonalization of a normal matrix, there are a unitary matrix $U$ and a diagonal matrix $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_N)$ such thatThe matrix 2-norm is invariant under multiplication by unitary matrices, soFor a diagonal matrix,with equality on a coordinate vector belonging to an eigenvalue of greatest modulus. Hencethe matrix 2-norm of a normal matrix.

The adjoint matrix of $A$ is its conjugate transpose:

A matrix is non-normal when it does not commute with its adjoint matrix:Nonorthogonal decaying eigenmodes can interfere constructively and produce transient growth. For example,has two negative eigenvalues but is non-normal. Starting from $x(0)=(0,1{)}^T$ givesAt $t=\log 2$, its squared Euclidean norm is $2.5^2+0.25^2>1=|x(0){|}^2$. The energy grows transiently even though both eigenmodes eventually decay.