Part 3 : Types of Matrices, Diagonal, and Transpose

Square Matrices

Matrices where (number of rows) = (number of columns)

A square matrix of order 3 × 3

For the matrices with whose number of rows and columns are unequal, we call them rectangular matrices.

Diagonal of a matrix

The elements on positions where (number of rows) = (number of columns) like a11, a22, a33 and so on, form diagonal of a matrix.

The highlighted elements make the diagonal of matrix

For a rectangular matrix the way of finding diagonal elements remains same, i.e. number or rows and columns should be equal, as shown below.

2 is not part of diagonal because its position is row 3 and column 4 (3 ≠ 4)

Identity or Unity Matrices

A square matrix with 1's as diagonal elements and 0’s as other elements is called an Identity matrix. It is denoted by I.

Identity matrices can be of any order, they just have to be square

As the name suggests, Identity matrix works like an identity, like 1 is identity in decimal number system (Any number, multiplied with 1 returns itself).

Hence, any matrix multiplied with suitable Identity matrix (conditions for matrix multiplication should be taken in account) returns the matrix itself.

matrix A × I = matrix A

If matrix A is of order 4 × 3 then it has to be multiplied with Identity matrix of order 3 × 3, denoted as I₃₃ or just I₃.

Zero or Null Matrices

Just like we have 0 in decimal number system, which on multiplication with any number returns 0 as product. We have Zero matrix which on multiplication with any matrix (satisfying conditions for matrix multiplication) returns a Zero matrix.

matrix A × Zero matrix = Zero matrix

A zero matrix of order 2 × 2

Unlike Identity matrices, Zero matrices can be rectangular.

Triangular Matrices

Upper Triangular Matrices

Matrices which have non-zero elements in and above diagonal . We denote upper triangular matrices with U.

Highlighting non-zero elements shows a triangle pointing upwards

Lower Triangular Matrices

Matrices which have non-zero elements in and below diagonal. We denote lower triangular matrices with L.

Highlighting non-zero elements shows a triangle pointing downwards

Transpose

Transpose of a matrix can be found by changing all the rows into columns or vice versa.

For example, element at position a12 (row 1 and column 2) will now be shifted to position a21 (row 2 and column 1), a13 to a31, a21 to a12 and so on.

Initial Matrix

Transposed Matrix

A transpose will be denoted by original matrix with “T” in superscript, like Aᵀ.

When we take transpose, only the diagonal elements don’t change place.

Classification of matrices on the basis of transpose

Symmetric Matrices

Matrices that remain unchanged on transposition.

Matrix H, on taking transpose remains unchanged.

Skew-symmetric Matrices

Matrices that on taking transpose become equal to their product with (-1) (scalar multiplication).

i.e. matrix Aᵀ = (-1) × matrix A

Taking transpose of matrix Y is same as multiplying it with (-1)

Notice that, the diagonal of skew-symmetric matrix consist of just 0’s because 0 is the only number which, on multiplication with (-1) returns itself (diagonal elements do not change place on taking transpose).

Read Part 4B : Tensors, Scalars, Vectors, and Matrices

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