Scalar Matrix

The scalar matrix is a square matrix having a constant value for all the elements of the principal diagonal, and the other elements of the matrix are zero. The scalar matrix is obtained by the product of the identity matrix with a numeric constant value.

Let us learn more about the definition of the scalar matrix, terms related to the scalar matrix, arithmetic operations of scalar matrix, and the examples of scalar matrix.

The scalar matrix is a square matrix having a constant value as every element of its principal diagonal, and all other elements are equal to zero. The scalar matrix is a square matrix having an equal number of rows and columns.

A = \( \begin{bmatrix}a&0&0\\0&a&0\\0&0&a\end{bmatrix}\)

Here in the above matrix the principal diagonal elements are all equal to the same numeric value of 'a', and all other elements of the matrix are equal to zero.

The scalar matrix is derived from an identity matrix, where the product of the identity matrix with a constant value, gives the scalar matrix. Here the constant 'k' is multiplied with an identity matrix to obtain the scalar matrix, and the order of the matrix is 3 × 3.

The following are some of the important terms, which are helpful in a better understanding of the concept of scalar matrix.

Identity Matrix

The identity matrix is a square matrix and is a multiplicative identity for matrices. The identity matrix contains 1 as its diagonal element and all other elements are equal to zero. Identity matrix has numerous applications in the multiplication of matrices, and in finding the inverse of a matrix, The identity matrix on multiplying with a constant value results in a scalar matrix.

I = \( \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

Diagonal Matrix

The diagonal matrix is also a square matrix, having elements of different value across the principal diagonal and all other elements are equal to zero. Further, if the diagonal elements of the diagonal matrix are all made equal, then it is called a scalar matrix.

D = \( \begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}\)

Principal Diagonal

The elements from the first element of the first row, to the last element of the last row, if connected with a straight line, all the elements falling on this imaginary straight line in the matrix, represent the principal diagonal. In an identity matrix, the principal diagonal elements are all equal to 1, and in a scalar matrix, all the principal diagonal elements are equal to a constant value.

Constant

This is a simple numeric value, which can be an integer, rational number, decimal number, or root value. The identity matrix is multiplied by a constant value to obtain the scalar matrix. A matrix multiplied by a constant value, multiplies with each of the elements of the matrix.

The matrix operations involving the scalar matrix are almost the same as the arithmetic operations of any other type of matrices. The addition and subtraction across a scalar matrix and any other matrix are the same as that across any two other matrices. But the multiplication of a scalar matrix with another matrix can be observed in the following few steps.

Scalar Matrix A = \( \begin{bmatrix}α&0&0\\0&α&0\\0&0&α\end{bmatrix}\), Matrix B = \( \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\).

A × B = \( \begin{bmatrix}α&0&0\\0&α&0\\0&0&α\end{bmatrix}\) × \( \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\)

= α\( \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\) × \( \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\)

= α\(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\)

A × B = \(\begin{bmatrix}αa&αb&αc\\αd&αe&αf\\αg&αh&αi\end{bmatrix}\)

A × B = αB

Thus the multiplication of a scalar matrix with any other matrix is equal to the multiplication of the constant element of the scalar matrix with all the elements of the other matrix.

Related Topics

The following topics help in a better understanding of scalar matrix.

• Matrices
• Square Matrix
• Transpose of a Matrix
• Adjoint of a Matrix
• Cofactor Formula
• Inverse of a Matrix

What Is a Scalar Matrix?

The scalar matrix is a square matrix having a constant value as every element of its principal diagonal, and all other elements are equal to zero. The scalar matrix is a square matrix having an equal number of rows and columns. An example of a scalar matrix is A = \( \begin{bmatrix}5&0&0\\0&5&0\\0&0&5\end{bmatrix}\)

Why Is It called a Scalar Matrix?

The scalar matrix is referred to by the name of scalar, as it is obtained by multiplying a scalar constant with the identity matrix. Scalar Constant × Identity Matrix = Scalar Matrix. Here 7 × \( \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\) = \( \begin{bmatrix}7&0&0\\0&7&0\\0&0&7\end{bmatrix}\) , which is a scalar matrix.

What Is the Order of a Scalar Matrix?

The scalar matrix is a square matrix, and it has an order n × n. The scalar matrix has an equal number of rows and columns.

Is a Zero Matrix Is a Scalar Matrix?

A zero matrix can be a called a scalar matrix. A zero matrix is a square matrix and all the principal diagonal elements are equal to a constant value, which is a zero. Hence a zero matrix can be called a scalar matrix.

What Is the Difference Between Scalar Matrix and Diagonal Matrix?

The only difference between the scalar matrix and a diagonal matrix is the elements of the principal diagonal. In a scalar matrix, the elements of the principal diagonal are all equal to the same constant value, and in a diagonal matrix the principal diagonal elements are all of different values. The non-diagonal elements in both the scalar matrix and diagonal matrix are all equal to zero. Scalar Matrix = \( \begin{bmatrix}a&0&0\\0&a&0\\0&0&a\end{bmatrix}\), and Diagonal Matrix = \( \begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}\).

How Can We Find the Scalar Matrix from Identity Matrix?

The scalar matrix can be obtained from the identity matrix by multiplying the identity matrix with a constant value. Constant Vaue × Identity Matrix = Scalar Matrix. K × \( \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\) = \( \begin{bmatrix}K&0&0\\0&K&0\\0&0&K\end{bmatrix}\)