The Diagonal Matrix

Matric is a highly useful tool for algebra and geometry. In geometry, they assist in representing various geometric objects and their properties. There are many types of matrices. A few matrix types are diagonal matrix, scalar matrix, anti-diagonal matrix, etc. All of these matrix forms are based on the diagonal of the matrix, which is why all these matrices are types of square matrices. A diagonal matrix is the type of matrix in which diagonal elements of the matrix numbers are non-zero while the other elements are zero. Let’s study in-depth diagonal matrices.

Types of matrices

Before we learn about diagrams with diagonals, let’s learn about a few matrices and a few concepts that we need to know to understand diagonal matrices well.

The following are the types of matrices and concepts.

• Square matrix:

A square matrix is a type of matrix where the number of rows is equal to the number of columns. The order of the square matrix is n × n.

A diagonal matrix is a one type of square matrix.

• Diagonal of the matrix:

Square types of matrices have diagonal elements. These are the elements of I × j, where the value of i is equal to the value of j. For example, the first element in the first row of the first column and the third element in the third row of the third column.

• Upper triangular matrix:

The upper triangular matrix is a matrix in which all the elements below the main diagonal are zero. It is a square matrix. An example of an upper triangular matrix is

• Lower triangle matrix:

The lower triangle matrix is a matrix in which all the elements above the main diagonal of the matrix are zero. It is a Square matrix. An example of a lower triangle matrix is

Now that we know all the necessary concepts, let’s learn about the diagonal matrix.

Diagonal matrix

A diagonal matrix is a square matrix in which all elements except the main diagonal elements are equal to zero. Diagonal elements can be any real or complex number, symbol and expression.

A diagonal matrix combines an upper triangular matrix and a lower triangle matrix. A diagonal matrix example  is given below.

The above matrix is the square matrix of order 3 × 3. It contains non-zero diagonal elements, and all other matrix elements are equal to zero. Therefore, the upper matrix is an example of a diagonal matrix. Various diagonal matrix calculators are available on the internet.

Let’s learn the features of the diagonal matrix.

Features of the diagonal matrix

Given below are a few features of the diagonal matrix.

• Each diagonal matrix is always a square matrix.

• The diagonal matrix can be divided into many types, such as identity matrix, null matrix, scalar matrix, etc. These matrices  have all the other elements except the diagonal element as zero.

• If a diagonal matrix is added to another diagonal matrix, the result will also be a diagonal matrix.

• Suppose two diagonal matrices of the same order are multiplied, then the resulting matrix elements are the products of the corresponding elements of both matrices.

• The addition and duplication of diagonal matrices follow the commutative law.

Let’s study the determinant of the diagonal matrix.

Determination of a diagonal matrix

Only square matrices can have diagonal elements and determinant values as the number of rows equals the number of columns.

The matrix determinant  is a single numerical value representing all matrix elements. The determinant of the diagonal matrix can be easily calculated by multiplying all the diagonal elements. Let’s take an example. Consider the matrix given below.

The above matrix is 3 × 3 and is only diagonal as non-zero numbers.

Let’s find out the determinant of the above matrix. The determinant of the diagonal matrix is equal to the product of its diagonal elements.

Since the diagonal elements of the upper matrix are 3, 6 and 9, their product equals 162, and the determinant value of the upper matrix will be equal to 162.

Hence, 162 is the determinant of a diagonal matrix.