Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics PDF Download

Symmetric Matrix

A real square matrix A = [a_ij] is called symmetric if transposition leaves it unchanged,

AT = A thus a_ij = a_ji .
The Eigen-values of symmetric matrix are always real.
Example:

Skew-Symmetric Matrix

A real square matrix A = [aij] is called skew-symmetric if transposition gives the negative of A,
AT = - A    thus aij = -a_ji
Every skew-symmetric matrix has all main diagonal entries zero.
The Eigen-values of skew-symmetric matrix are pure imaginary or zero.
Example :

NOTE:
Any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S, where

Example:

Orthogonal Matrix

A real square matrix A =[a_ij] is called orthogonal if transposition gives the inverse of A,

AT = A^-1

NOTE:

(i) A real square matrix is orthogonal if and only if its column vectors and also its row vectors form an orthonormal system, that is

Thus A^TA = I

(ii) The determinate of an orthogonal matrix has the value +1 or -1.

(iii) The eigenvalues of an orthogonal matrix A are real or complex conjugate in pairs and have absolute value 1.
Example:
Its characteristic equation is 

Hermitian, Skew-Hermitian, and Unitary Matrices (Complex Matrices)

The complex conjugate of an matrix A is formed by taking the complex conjugate of each element. Thus
For the conjugate transpose, we use the notation
Example:

Hermitian Matrix

A square matrix A = [a_ij] is called Hermitian if

If A is Hermitian, the entries on the main diagonal must satisfy  that is they are real.

If a Hermitian matrix is real, then  = A^T = A. Hence a real Hermitian matrix is a symmetric matrix.

The eigenvalues of a Hermitian matrix (and thus a symmetric matrix) are real.
Example: The eigenvalues are 9, 2 .

Skew- Hermitian Matrix

A square matrix A = [aij] is called skew-Hermitian if

• If A is skew-Hermitian, then entries on the main diagonal must satisfy  hence ajj must be pure imaginary or 0.
• If a skew-Hermitian matrix is real, then  Hence a real skew-Hermitian matrix is a skew-symmetric matrix.
• The eigenvalues of a skew-Hermitian matrix (and thus a skew-symmetric matrix) are pure imaginary or 0.

Example:
The eigenvalues are 4i, - 2i.

Unitary Matrix

A square matrix A = [a_ij] is called unitary if

• If a unitary matrix is real, then  Hence a real unitary matrix is an orthogonal matrix.
• The eigenvalues of a unitary matrix (and thus an orthogonal matrix) have absolute value 1.

Example:
The eigenvalues are

Similarity of Matrices, Basis of Eigenvectors, and Diagonalisation

Eigenvectors of an n x n matrix A may (or may not) form a basis. If they do, we can use them for “diagonalizing” A , that is, for transforming it into diagonal form with the eigenvalues on the main diagonal.

Similarity of Matrices

An n x n matrix  is called similar to an n x n matrix A if

 = P^-1AP

For some (nonsingular) n x n matrix P . This transformation, which gives  from A , is called similarity transformation.

• If  is similar to A, then  has the same eigenvalues as A .
• If x is an eigenvector of A, then y = P^-1x is an eigenvector of  corresponding to same eigenvalue.
• If λ₁,λ₂,....λ_k be distinct eigenvalues of an n x n matrix. Then corresponding eigenvectors x₁,x₂,....x_k form linearly independent set.

Basis of Eigenvectors

If an n x n matrix A has n distinct eigenvalues, then A has a basis of eigenvectors

• A Hermitian, skew-Hermitian or unitary matrix has a basis of eigenvectors.
• A symmetric matrix has an orthonormal basis of eigenvectors.

Diagonalisation

If an n x n matrix A has a basis of eigenvectors, then

D = X^-1 AX
is diagonal, with the eigenvalues of A as the entries on the main diagonal. Here X is the matrix with these eigenvectors as column vectors. Also

D^m = X^-1 A^mX

A square matrix which is not diagonalizable is called defective.

Example: A = has eigenvalues 6, 1. The corresponding eigenvectors are

Thus

Example: A = has eigenvalues λ₁ =3,λ₂,=2,λ₁ =1.
The eigenvectors of A corresponds to eigen value respectively λ₁ = 3. λ₂ = 2. λ₁ = 1 are

Now, let X be the matrix with these eigenvectors as its columns:

Note there is no preferred order of the eigenvectors inX; changing the order of the eigenvectors in X just changes the order of the eigenvalues in the diagonalzed form of A.
Thus, D = X^-1 AX =
Note that the eigenvalues λ₁, = 3, λ₂ = 2, λ₁ = 1 appear in the diagonal matrix.
Functional Matrices

If the matrix A is diagonalizable, then we can find a matrix X and a diagonal matrix D such that

D = X^-1 AX ⇒ A = XDX^-1 and Aⁿ = XDⁿX^-1
Applying the power series definition to this decomposition, we find that f (A) is defined

by

f (A) = I + α A + βA² + y A³ +     (a, β,y are coefficient of Taylor Expansion)

⇒ f (A) = XIX^-1 + aXDX^-1 + βXD²X^-1 + γXD³X^-1 +.......     ∵ Aⁿ = XDⁿX^-1

⇒ f (A) = X [I + αD + βD² + yD³ + ........] X^-1

where d₁, d₂, d_n denote the diagonal entries of D.
Note: If A is itself diagonal then f (A) = f (D)
Example 10: If matrix

For matrix A Eigenvalues are λ₁,λ₂ and Eigenvectors are respectively

Thus

Example: If matrix
then
Example 11: If matrix A = then find e^A .

For eigenvalues (A - λI) = 0 ⇒
Eigenvector can be determined by the equation AX = λX .

For λ₁ = 1

Normalized eigenvector can be determined by relation X₁^TX₁ = 1

Normalized Eigenvector corresponds to λ₁ = 1 is
For λ₂ = -1

Normalized eigenvector can be determined by relation X₂^TX₂ = 1.

Normalized Eigenvector corresponds to λ₂ = -1 is
Hence,  The cofactor of X is X^c =

Hence 

NOTE: Always try to express X as unitary matrix because its inverse is same. If X is not unitary matrix then we have to find its inverse.

Thus,