Matrix in Quantum Mechanics

Matrix:

A matrix is a rectangular array of real numbers containing m rows and n columns. An mxn matrix contains m rows and n columns and can be expressed as:

The quantities a_ij are called the entries or components of the matrix. The entry at the ith row and jth column of matrix A is denoted by the corresponding lowercase alphabet with subscripts.

Square Matrix:

A square matrix is a matrix in which the number of rows is equal to the number of columns, that is m = n. An n x n matrix is known as a square matrix of order n. Square matrices have some unique properties such as symmetry and anti-symmetry. Furthermore, finding determinants and computing eigenvalues are only possible in square matrices.

Diagonal (Triangular) Matrix:

A square matrix of order n is called a diagonal matrix if

a_ii = d_i

a_ij = 0 if i≠j

For example, consider the 3 x 3 matrix shown below. The elements are a₁₁ = 8, a₂₂ = 10, a₃₃ = 3. All other entries above and below the diagonal line are 0s. Hence, this is a diagonal matrix.

Symmetric Matrix:

A square matrix is symmetry if it is equal to its transpose. In other words, an n x n matrix A is symmetric if A = A^{T such a matrix is symmetric about its main diagonal. Some of the important properties of symmetric matrices:}

1. The transpose of a symmetric matrix A is also symmetric.

2. The product of a symmetric matrix A and a scalar c that is cA is also symmetric.

3. The inverse of the transpose of a symmetric matrix A is equal to the inverse of the matrix.

(A^T)’ = A’

4. If A and B are two symmetric matrices and (A+B)^T = A^T + B^T = A + B then (A + B) is a symmetric matrix.

5. The product of two symmetric matrices is also symmetric if the two matrices commute, that is AB = BA.

Transpose Matrix:

The transpose of m x n matrix A denoted by A^T, is an n x m matrix with entries.

(A^T)_ij = a_ji

Thus, the transpose of a matrix can be derived by replacing an ith row of the matrix of A with the ith column of A. The transpose operation is –

Orthogonal Matrix:

A square matrix is called orthogonal if the transpose of the matrix A is equal to its inverse:

A^T = A^-1