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1 Mathematical Physics
Hermetian matrix
If A = A, so that a i j = a ji for all values of i and j. Diagonal elements of an
Hermitian matrix are real numbers.
Orthogonal matrix
A square matrix is said to be orthogonal if A A
= A
A = I , i.e. A
= A
−1
The column vector (row vectors) of an orthogonal matrix A are mutually orthogonal unit vectors.
The inverse and the transpose of an orthogonal matrix are orthogonal.
The product of any two or more orthogonal matrices is orthogonal.
The determinant of an Orthogonal matrix is ±1.
Unitary matrix
A square matrix is called a unitary matrix if (A)
A = A( A)
= I , i.e. if ( A)
= A
−1 .
The column vectors (row vectors) of an n-square unitary matrix are an orthonormal set.
The inverse and the transpose of a unitary matrix are unitary.
The product of two or more unitary matrices is unitary.
The determinant of a unitary matrix has absolute value 1.
Unitary transformations
The linear transformation Y = AX (where A is unitary and X is a vector), is called
a unitary transformation.
If the matrix is unitary, the linear transformation preserves length.
Rank of a matrix
If |A| | = 0, it is called non-singular; if |A| = 0, it is called singular.
A non-singular matrix is said to have rank r if at least one of its r -square minors
is non-zero while if every (r + 1) minor, if it exists, is zero.
Elementary transformations
(i) The interchange of the ith rows and jth rows or ith column or jth column.
(ii) The multiplication of every element of the ith row or ith column by a non-zero
scalar.
(iii) The addition to the elements of the ith row (column) by k (a scalar) times the
corresponding elements of the jth row (column). These elementary transformations known as row elementary or row transformations do not change the
order of the matrix.
1 Mathematical Physics
Hermetian matrix
If A = A, so that a i j = a ji for all values of i and j. Diagonal elements of an
Hermitian matrix are real numbers.
Orthogonal matrix
A square matrix is said to be orthogonal if A A
= A
A = I , i.e. A
= A
−1
The column vector (row vectors) of an orthogonal matrix A are mutually orthogonal unit vectors.
The inverse and the transpose of an orthogonal matrix are orthogonal.
The product of any two or more orthogonal matrices is orthogonal.
The determinant of an Orthogonal matrix is ±1.
Unitary matrix
A square matrix is called a unitary matrix if (A)
A = A( A)
= I , i.e. if ( A)
= A
−1 .
The column vectors (row vectors) of an n-square unitary matrix are an orthonormal set.
The inverse and the transpose of a unitary matrix are unitary.
The product of two or more unitary matrices is unitary.
The determinant of a unitary matrix has absolute value 1.
Unitary transformations
The linear transformation Y = AX (where A is unitary and X is a vector), is called
a unitary transformation.
If the matrix is unitary, the linear transformation preserves length.
Rank of a matrix
If |A| | = 0, it is called non-singular; if |A| = 0, it is called singular.
A non-singular matrix is said to have rank r if at least one of its r -square minors
is non-zero while if every (r + 1) minor, if it exists, is zero.
Elementary transformations
(i) The interchange of the ith rows and jth rows or ith column or jth column.
(ii) The multiplication of every element of the ith row or ith column by a non-zero
scalar.
(iii) The addition to the elements of the ith row (column) by k (a scalar) times the
corresponding elements of the jth row (column). These elementary transformations known as row elementary or row transformations do not change the
order of the matrix.