Matrix in Quantum Mechanics
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 aij 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.
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
aii = di
aij = 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.
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 = AT 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.
(AT)’ = A’
4. If A and B are two symmetric matrices and (A+B)T = AT + BT = 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.
The transpose of m x n matrix A denoted by AT, is an n x m matrix with entries.
(AT)ij = aji
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 –
A square matrix is called orthogonal if the transpose of the matrix A is equal to its inverse:
AT = A-1