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

Symmetric Matrix

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

AT = A thus aij = aji .
The Eigen-values of symmetric matrix are always real.
Example: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

Skew-Symmetric Matrix

A real square matrix A = [aij] is called skew-symmetric if transposition gives the negative of A,
AT = - A    thus aij = -aji
Every skew-symmetric matrix has all main diagonal entries zero.
The Eigen-values of skew-symmetric matrix are pure imaginary or zero.
Example : Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

NOTE:
Any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S, where
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Example: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

Orthogonal Matrix

A real square matrix A =[aij] 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
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Thus ATA = 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: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Its characteristic equation is Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

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 Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
For the conjugate transpose, we use the notation Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Example: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

Hermitian Matrix

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

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

If A is Hermitian, the entries on the main diagonal must satisfy Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics that is they are real.

If a Hermitian matrix is real, then Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics = AT = A. Hence a real Hermitian matrix is a symmetric matrix.

The eigenvalues of a Hermitian matrix (and thus a symmetric matrix) are real.
Example: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - PhysicsThe eigenvalues are 9, 2 .

Skew- Hermitian Matrix

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

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

  • If A is skew-Hermitian, then entries on the main diagonal must satisfy Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - PhysicsSymmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics hence ajj must be pure imaginary or 0.
  • If a skew-Hermitian matrix is real, then Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics 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:Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
The eigenvalues are 4i, - 2i.

Unitary Matrix

A square matrix A = [aij] is called unitary if
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

  • If a unitary matrix is real, then Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics 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: Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
The eigenvalues are Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

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 λ12,....λk be distinct eigenvalues of an n x n matrix. Then corresponding eigenvectors x1,x2,....xk 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= X-1 AmX

A square matrix which is not diagonalizable is called defective.

Example: A = Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physicshas eigenvalues 6, 1. The corresponding eigenvectors are Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

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

Example: A = Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physicshas eigenvalues λ1 =3,λ2,=2,λ1 =1.
The eigenvectors of A corresponds to eigen value respectively λ1 = 3. λ2 = 2. λ1 = 1 are
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

Now, let X be the matrix with these eigenvectors as its columns:
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
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 = Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
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
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
where d1, d2, dn denote the diagonal entries of D.
Note: If A is itself diagonal then f (A) = f (D)
Example 10: If matrix Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

For matrix A Eigenvalues are λ12 and Eigenvectors are respectively Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

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

Example: If matrix Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
then Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Example 11: If matrix A = Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physicsthen find eA .

For eigenvalues (A - λI) = 0 ⇒ Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Eigenvector can be determined by the equation AX = λX .
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
For λ1 = 1
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Normalized eigenvector can be determined by relation X1TX1 = 1
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Normalized Eigenvector corresponds to λ1 = 1 is Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
For λ2 = -1
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Normalized eigenvector can be determined by relation X2TX2 = 1.
Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Normalized Eigenvector corresponds to λ2 = -1 is Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics
Hence, Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics The cofactor of X is Xc = Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices | Mathematical Methods - Physics

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

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

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

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FAQs on Symmetric, Skew-Symmetric, Orthogonal & Complex Matrices - Mathematical Methods - Physics

1. What is the difference between a Hermitian and a Skew-Hermitian matrix?
Ans. A Hermitian matrix is a complex matrix that is equal to its conjugate transpose, meaning that the matrix is equal to the conjugate transpose of itself. On the other hand, a Skew-Hermitian matrix is a complex matrix that is equal to the negative of its conjugate transpose, meaning that the matrix is equal to the negative of the conjugate transpose of itself.
2. What are the properties of a Unitary matrix?
Ans. A Unitary matrix is a complex square matrix that satisfies the following properties: - The matrix multiplied by its conjugate transpose gives the identity matrix: A * A^H = I, where A^H represents the conjugate transpose of matrix A. - The columns of the matrix are orthonormal, meaning that the dot product of any two columns is zero if the columns are different and one if the columns are the same.
3. How do you determine if two matrices are similar?
Ans. Two matrices A and B are considered similar if there exists an invertible matrix P such that P^(-1)AP = B. In other words, if there is a matrix P that can transform matrix A into matrix B by similarity transformation, then A and B are similar.
4. What is the significance of a basis of eigenvectors in diagonalizing a matrix?
Ans. A basis of eigenvectors is crucial in diagonalizing a matrix. Diagonalization is the process of transforming a matrix into a diagonal matrix by similarity transformation. The basis of eigenvectors forms the columns of the matrix P in the equation P^(-1)AP = D, where D is the diagonal matrix. These eigenvectors provide a convenient basis in which the matrix can be represented in a simpler form, making it easier to analyze and compute properties of the matrix.
5. What are the properties of Symmetric, Skew-Symmetric, Orthogonal, and Complex matrices?
Ans. - Symmetric Matrix: A symmetric matrix is a square matrix that is equal to its transpose. In other words, for a matrix A, A = A^T. - Skew-Symmetric Matrix: A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose. In other words, for a matrix A, A = -A^T. - Orthogonal Matrix: An orthogonal matrix is a square matrix whose columns and rows are orthonormal. This means that the dot product of any two different columns or rows is zero, and the dot product of a column or row with itself is one. Additionally, the product of an orthogonal matrix and its transpose is equal to the identity matrix. - Complex Matrix: A complex matrix is a matrix with complex entries. The entries of a complex matrix can be any complex numbers, which consist of a real part and an imaginary part.
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