Chapter 56. Hermitian and Normal Matrices

Hermitian and normal matrices are central objects in complex linear algebra. Hermitian matrices are the complex analogue of real symmetric matrices. Normal matrices form a broader class that includes Hermitian, unitary, and skew-Hermitian matrices.

Their importance comes from the spectral theorem. Hermitian matrices can be diagonalized by unitary matrices with real eigenvalues. More generally, every normal matrix can be unitarily diagonalized. This means that after a change to orthonormal coordinates, the matrix becomes diagonal. The geometry of the space is preserved because the change of basis is unitary. (en.wikipedia.org)

56.1 Hermitian Matrices

A complex square matrix $A\in\mathbb{C}^{n\times n}$ is Hermitian if

A^*=A,

where

A^*=\overline{A}^{\,T}

is the conjugate transpose.

Entrywise, this means

a_{ij}=\overline{a_{ji}}

for all $i,j$ .

For example,

A= \begin{bmatrix} 2 & 1+i\\ 1-i & 3 \end{bmatrix}

is Hermitian because

\overline{1+i}=1-i.

The diagonal entries of a Hermitian matrix are always real. Indeed,

a_{ii}=\overline{a_{ii}},

which implies

a_{ii}\in\mathbb{R}.

Hermitian matrices are the complex analogue of symmetric matrices:

Real case	Complex case
$A^T=A$	$A^*=A$
Symmetric	Hermitian

56.2 Quadratic Forms

For a Hermitian matrix $A$ , the quantity

x^*Ax

is always real.

Indeed,

\overline{x^*Ax} = (x^*Ax)^* = x^*A^*x.

Since $A=A^*$ ,

x^*A^*x=x^*Ax.

Thus

\overline{x^*Ax}=x^*Ax,

so $x^*Ax\in\mathbb{R}$ .

This property is fundamental. Hermitian matrices represent real-valued quadratic energy expressions in complex coordinates.

56.3 Positive Definite Hermitian Matrices

A Hermitian matrix $A$ is positive definite if

x^*Ax>0 \quad \text{for all nonzero }x\in\mathbb{C}^n.

It is positive semidefinite if

x^*Ax\ge 0

for all $x$ .

For example,

A= \begin{bmatrix} 2&0\\ 0&3 \end{bmatrix}

is positive definite because

x^*Ax = 2|x_1|^2+3|x_2|^2

is strictly positive for every nonzero vector.

Positive definite Hermitian matrices define Hermitian inner products:

\langle x,y\rangle_A=x^*Ay.

They appear throughout optimization, statistics, differential equations, and numerical linear algebra.

56.4 Eigenvalues of Hermitian Matrices

Hermitian matrices have real eigenvalues.

Suppose

Av=\lambda v

for some nonzero vector $v$ . Then

v^*Av=v^*(\lambda v).

Using scalar multiplication,

v^*Av=\lambda v^*v.

Since $A$ is Hermitian, the quantity $v^*Av$ is real. Also,

v^*v>0.

Therefore

\lambda = \frac{v^*Av}{v^*v}

is real.

Thus every eigenvalue of a Hermitian matrix lies in $\mathbb{R}$ . This is one of the most important structural properties of Hermitian matrices. (mathworld.wolfram.com)

56.5 Orthogonality of Eigenvectors

Eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal.

Suppose

Av=\lambda v, \qquad Aw=\mu w,

with

\lambda\ne\mu.

Compute

v^*Aw.

Since $Aw=\mu w$ ,

v^*Aw=\mu v^*w.

On the other hand,

v^*Aw=(A^*v)^*w.

Because $A=A^*$ ,

A^*v=Av=\lambda v.

Thus

v^*Aw = (\lambda v)^*w = \overline{\lambda}v^*w.

Since $\lambda$ is real,

\overline{\lambda}=\lambda.

Therefore

\lambda v^*w=\mu v^*w.

Hence

(\lambda-\mu)v^*w=0.

Since $\lambda\ne\mu$ ,

v^*w=0.

Thus $v\perp w$ .

56.6 Spectral Theorem for Hermitian Matrices

Every Hermitian matrix can be unitarily diagonalized.

More precisely, if $A\in\mathbb{C}^{n\times n}$ is Hermitian, then there exists a unitary matrix $U$ and a real diagonal matrix $\Lambda$ such that

A=U\Lambda U^*.

Equivalently,

U^*AU=\Lambda.

The columns of $U$ are orthonormal eigenvectors of $A$ , and the diagonal entries of $\Lambda$ are the eigenvalues of $A$ .

This is the complex spectral theorem for Hermitian matrices. It is one of the central theorems of linear algebra. Hermitian matrices admit an orthonormal eigenbasis and are therefore unitarily diagonalizable. (mathworld.wolfram.com)

56.7 Example of Hermitian Diagonalization

Consider

A= \begin{bmatrix} 2&1\\ 1&2 \end{bmatrix}.

This matrix is Hermitian because it is real symmetric.

The characteristic polynomial is

\det(A-\lambda I) = \begin{vmatrix} 2-\lambda & 1\\ 1 & 2-\lambda \end{vmatrix} = (2-\lambda)^2-1.

Thus

\lambda^2-4\lambda+3=0.

The eigenvalues are

\lambda_1=3, \qquad \lambda_2=1.

For $\lambda_1=3$ ,

(A-3I)v=0

gives eigenvectors proportional to

\begin{bmatrix} 1\\ 1 \end{bmatrix}.

For $\lambda_2=1$ ,

(A-I)v=0

gives eigenvectors proportional to

\begin{bmatrix} 1\\ -1 \end{bmatrix}.

Normalize them:

u_1= \frac{1}{\sqrt2} \begin{bmatrix} 1\\ 1 \end{bmatrix}, \qquad u_2= \frac{1}{\sqrt2} \begin{bmatrix} 1\\ -1 \end{bmatrix}.

Then

U= \frac{1}{\sqrt2} \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix}

is unitary, and

A = U \begin{bmatrix} 3&0\\ 0&1 \end{bmatrix} U^*.

56.8 Normal Matrices

A matrix $A\in\mathbb{C}^{n\times n}$ is normal if

A^*A=AA^*.

This condition means that $A$ commutes with its adjoint.

Every Hermitian matrix is normal because

A^*=A

implies

A^*A=A^2=AA^*.

Every unitary matrix is normal because

A^*A=I=AA^*.

Every skew-Hermitian matrix is also normal.

The class of normal matrices includes several important families:

Matrix type	Defining condition
Hermitian	$A^*=A$
Skew-Hermitian	$A^*=-A$
Unitary	$A^*A=I$
Normal	$A^A=AA^$

56.9 Skew-Hermitian Matrices

A matrix $A$ is skew-Hermitian if

A^*=-A.

Its diagonal entries are purely imaginary or zero, because

a_{ii}=-\overline{a_{ii}}.

Av=\lambda v,

then

v^*Av=\lambda v^*v.

But

(v^*Av)^* = v^*A^*v = -v^*Av.

Thus

v^*Av

is purely imaginary, so $\lambda$ is purely imaginary.

Skew-Hermitian matrices are the complex analogue of skew-symmetric real matrices.

56.10 Spectral Theorem for Normal Matrices

A matrix $A$ is normal if and only if it is unitarily diagonalizable.

That is,

A^*A=AA^*

if and only if there exists a unitary matrix $U$ and a diagonal matrix $\Lambda$ such that

A=U\Lambda U^*.

Unlike the Hermitian case, the diagonal entries of $\Lambda$ need not be real.

Thus a normal matrix may have complex eigenvalues, but it still has an orthonormal eigenbasis.

This theorem characterizes normal matrices completely in terms of orthonormal diagonalization. A complex matrix is normal exactly when it can be diagonalized by a unitary similarity transformation. (en.wikipedia.org)

56.11 Example of a Normal Matrix

Consider

A= \begin{bmatrix} 1&0\\ 0&i \end{bmatrix}.

Then

A^*= \begin{bmatrix} 1&0\\ 0&-i \end{bmatrix}.

Compute

A^*A = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} = AA^*.

Thus $A$ is normal.

It is not Hermitian because

A^*\ne A.

Its eigenvalues are

1 \quad \text{and} \quad i.

The standard basis vectors form an orthonormal eigenbasis, so the matrix is already unitarily diagonalized.

56.12 Unitary Diagonalization

Suppose

A=U\Lambda U^*

with $U$ unitary and $\Lambda$ diagonal.

Then

A^* = (U\Lambda U^*)^* = U\Lambda^*U^*.

Thus

A^*A = U\Lambda^*\Lambda U^*,

and

AA^* = U\Lambda\Lambda^*U^*.

Since diagonal matrices commute,

\Lambda^*\Lambda = \Lambda\Lambda^*.

Therefore

A^*A=AA^*.

Thus every unitarily diagonalizable matrix is normal.

This proves one direction of the spectral theorem for normal matrices.

56.13 Real Symmetric Matrices

A real symmetric matrix is a Hermitian matrix with real entries.

Thus the complex spectral theorem contains the real symmetric spectral theorem as a special case.

A^T=A,

then there exists an orthogonal matrix $Q$ such that

A=Q\Lambda Q^T.

The eigenvalues are real, and the eigenvectors may be chosen orthonormal.

This is the real form of the spectral theorem.

56.14 Unitary Matrices and Eigenvalues

If $U$ is unitary and

Uv=\lambda v

for some nonzero vector $v$ , then

\|Uv\|_2=\|v\|_2.

But

Uv=\lambda v,

\|\lambda v\|_2=|\lambda|\|v\|_2.

Therefore

|\lambda|=1.

Thus every eigenvalue of a unitary matrix lies on the complex unit circle.

This geometric condition reflects norm preservation.

56.15 Normality and Orthogonality

For arbitrary matrices, eigenvectors corresponding to distinct eigenvalues need not be orthogonal.

For normal matrices, they are orthogonal.

Suppose

Av=\lambda v, \qquad Aw=\mu w,

with

\lambda\ne\mu.

Then

A^*v=\overline{\lambda}v

for eigenvectors of a normal matrix associated with a unitary diagonalization. Using this property and the commutation relation

A^*A=AA^*,

one proves

v^*w=0.

Thus normality is precisely the condition that allows diagonalization by an orthonormal basis.

56.16 Functional Calculus

A=U\Lambda U^*

is unitarily diagonalized, then functions of $A$ are computed by applying the function to the diagonal entries:

f(A)=Uf(\Lambda)U^*,

where

f(\Lambda) = \operatorname{diag}(f(\lambda_1),\ldots,f(\lambda_n)).

For example,

A^k = U\Lambda^kU^*,

and

e^A = Ue^\Lambda U^*.

This is important in differential equations, quantum mechanics, and matrix analysis.

56.17 Rayleigh Quotients

For a Hermitian matrix $A$ , the quantity

R_A(x)=\frac{x^*Ax}{x^*x}

is called the Rayleigh quotient.

Since $A$ is Hermitian,

R_A(x)\in\mathbb{R}.

If $x$ is an eigenvector with eigenvalue $\lambda$ , then

R_A(x)=\lambda.

The minimum and maximum eigenvalues satisfy

\lambda_{\min} \le R_A(x) \le \lambda_{\max}.

Rayleigh quotients connect eigenvalues with optimization.

56.18 Positive Semidefinite Decomposition

A Hermitian matrix $A$ is positive semidefinite if and only if all its eigenvalues are nonnegative.

Indeed, if

A=U\Lambda U^*,

then

x^*Ax = x^*U\Lambda U^*x.

Let

y=U^*x.

Since $U$ is unitary,

x^*Ax=y^*\Lambda y = \sum_{i=1}^n \lambda_i|y_i|^2.

This quantity is nonnegative for all $x$ exactly when

\lambda_i\ge 0

for all $i$ .

Thus positivity is determined entirely by the eigenvalues.

56.19 Numerical Importance

Hermitian and normal matrices are computationally important because their spectral structure is stable and geometrically well behaved.

Hermitian matrices have:

Property	Consequence
Real eigenvalues	Stable physical interpretation
Orthogonal eigenspaces	Stable decompositions
Unitary diagonalization	Numerically robust spectral methods
Positive definiteness	Stable optimization and factorization

Algorithms for Hermitian matrices are often faster and more stable than general matrix algorithms.

Examples include:

Problem	Specialized method
Hermitian eigenvalue problem	Symmetric QR, divide-and-conquer
Positive definite systems	Cholesky factorization
Singular values	Hermitian reductions
Optimization	Hessian analysis

Many scientific computing libraries exploit Hermitian structure explicitly.

56.20 Summary

A Hermitian matrix satisfies

A^*=A.

Its eigenvalues are real, and eigenvectors for distinct eigenvalues are orthogonal.

Every Hermitian matrix can be diagonalized by a unitary matrix:

A=U\Lambda U^*.

A matrix is normal if

A^*A=AA^*.

Normal matrices are exactly the matrices that admit unitary diagonalization.

Hermitian, skew-Hermitian, and unitary matrices are all normal. Their structure is governed by orthonormal eigenbases and the geometry of Hermitian inner products.

The spectral theorem for Hermitian and normal matrices is one of the central organizing principles of linear algebra.