Types of Matrices, Triangular matrix, Diagonal matrix, Scalar matrix, Identity matrix, Symmetric matrix, Skew-symmetric matrix, Direct Sum, Transpose of a matrix

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Types of matrices — triangular, diagonal, scalar, identity, symmetric, skew-symmetric, periodic, nilpotent

Upper triangular matrix. A square matrix in which all the elements below the diagonal are zero i.e. a matrix of type:

Lower triangular matrix. A square matrix in which all the elements above the diagonal are zero i.e. a matrix of type

Diagonal matrix. A square matrix in which all of the elements are zero except for the diagonal elements i.e. a matrix of type

It is often written as D = diag(a₁₁, a₂₂, a₃₃, ... , a_nn)

Scalar matrix. A diagonal matrix in which all of the diagonal elements are equal to some constant “k” i.e. a matrix of type

Identity matrix. A diagonal matrix in which all of the diagonal elements are equal to “1" i.e. a matrix of type

An identity matrix of order nxn is denoted by I_n.

Transpose of a matrix. The matrix resulting from interchanging the rows and columns in the given matrix. The transpose of

The first row of A becomes the first column of A^T, the second row of A becomes the second column of A^T, etc.. It corresponds to a “flip”of the matrix about the diagonal running down from the upper left corner.

Symmetric matrix. A square matrix in which corresponding elements with respect to the diagonal are equal; a matrix in which a_ij = a_ji where a_ij is the element in the i-th row and j-th column; a matrix which is equal to its transpose; a square matrix in which a flip about the diagonal leaves it unchanged. Example:

Skew-symmetric matrix. A square matrix in which corresponding elements with respect to the diagonal are negatives of each other; a matrix in which a_ij = -a_ji where a_ij is the element in the i-th row and j-th column; a matrix which is equal to the negative of its transpose. The diagonal elements are always zeros. Example:

Direct Sum. Let A₁, A₂, ... ,A_s be square matrices of respective orders m₁, m₂, ... ,m_s .

The generalization

of the diagonal matrix is called the direct sum of the A_i.

Inverse of a matrix. If A and B are square matrices such that AB = BA = I where I is the identity matrix , then B is called the inverse of A and we write B = A^-1 . The matrix B also has A as an inverse and we can write A = B^-1 .

Commutative and anti-commutative matrices. If A and B are square matrices such that AB = BA, then A and B are called commutative or are said to commute. If AB = -BA, the matrices are said to anti-commute.

Periodic matrix. A matrix A for which A^k+1 = A , where k is a positive integer. If k is the least positive integer for which A^k+1 = A , then A is said to be of period k. If k = 1, so that A² = A, then A is called idempotent.

Nilpotent matrix. A matrix A for which A^p = 0, where p is some positive integer. If p is the least positive integer for which A^p = 0, then A is said to be nilpotent of index p.

References.

Ayres. Matrices (Schaum).