Diagonal Matrix – Explanation & Examples

A diagonal matrix is a square matrix whose elements, other than the diagonal, are zero. There are certain conditions that must be met for a matrix to be called a diagonal matrix. Firstly, let’s check the formal definition of a diagonal matrix.

A square matrix in which all the elements except the principal diagonal are zero is known as a diagonal matrix.

In this article, we are going to take a close look at what makes a matrix diagonal, how to find diagonal matrices, properties of diagonal matrices, and the determinant of a diagonal matrix. Let’s start!

What is a Diagonal Matrix?

A matrix to be classified as a diagonal matrix, it has to meet the following conditions:

• square matrix
• all elements (entries) of the matrix, other than the principal diagonal, has to be $0$

A square matrix is said to be:

• lower triangular if its elements above the principal diagonal are all $0$
• upper triangular if its elements below the principal diagonal are all $0$

Lower Triangular Matrix

Upper Triangular Matrix

A diagonal matrix is a special square matrix that is BOTH upper and lower triangular since all elements, whether above or below the principal diagonal, are $0$.

How to find Diagonal Matrix

To find, or identify, a diagonal matrix, we need to see if it is a square matrix and all the elements besides the principal diagonal (diagonal that runs from top left to bottom right) are $0$. Let’s take a look at the matrices shown below:

$A=\left(\begin{array}{cc}3& 0\\ 0& –3\end{array}\right)$

$B=\left(\begin{array}{ccc}1& 0& 0\\ 0& 5& 0\\ 0& 0& –2\end{array}\right)$

$C=\left[\begin{array}{cc}10& 0\\ 0& 12\\ 0& 12\end{array}\right]$

$D=\left[\begin{array}{ccc}–5& 0& 0\\ 0& 0& 0\\ 0& 0& 9\end{array}\right]$

Note the following observations about each of the $4$ matrices shown above:

• Matrix $A$ is a square matrix because its has the same number of rows and columns ($2\times 2$ matrix). The principal diagonal has entries $3$ and $-3$, respectively. All other entries are $0$. Thus, this is a diagonal matrix.
• Matrix $B$ is a square matrix because its has the same number of rows and columns ($3\times 3$ matrix). The principal diagonal has entries $1$, $5$, and $-2$, respectively. All other entries are $0$. Thus, this is a diagonal matrix.
• A false glance at Matrix $C$ will make us think that it is a diagonal matrix. But, first and foremost, it is not a square matrix. Hence, it cannot be a diagonal matrix.
• This one’s a bit tricky. First of all, it is in fact a square matrix ( $3\times 3$ ). You will think that it is not a diagonal matrix.But why? Is it because there’s a $0$ in the middle?
  

  If you take a closer look at the definition of a diagonal matrix, you will see that nowhere does it say that the entries in the diagonal cannot be $0$! The condition is that the elements other than the diagonal has to be $0$. So, even if there are elements that are $0$ in the diagonal, it won’t matter. As long as the elements besides the diagonal are $0$, it will be a diagonal matrix.

  Thus, Matrix $D$ is in fact a diagonal matrix!

This brings us to $2$ special types of diagonal matrices:

• Identity Matrix
• Zero Matrix

Identity Matrix

This is a square matrix in which all the entries in the principal diagonal are $1$ and all other elements are $0$. A $2\times 2$ and a $3\times 3$ identity matrices are shown below.

$2\times 2$ identity matrix

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$3\times 3$ identity matrix

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

You can read more about identity matrices here.

Zero Matrix

A matrix in which all the elements are $0$.  A $2\times 2$ and a $3\times 3$ zero matrices are shown below.

$2\times 2$ zero matrix

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$3\times 3$ zero matrix

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Now, let’s look at some properties of diagonal matrices.

Properties of Diagonal Matrices

There are several properties of diagonal matrices but for the purpose of this article, we will look at $3$ properties of diagonal matrices. Below, we take a look at the properties and their examples.

Property 1:

When $2$ diagonal matrices of the same order are added or multiplied together, the resultant matrix is another diagonal matrix with the same order.

Consider the matrices shown below:

$A=\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]$
$B=\left[\begin{array}{cc}1& 0\\ 0& 5\end{array}\right]$

Now, we add both the $2\times 2$ matrices and show that the resultant matrix is also diagonal.

$A+B=\left[\begin{array}{cc}3+1& 0+0\\ 0+0& 4+5\end{array}\right]$

$A+B=\left[\begin{array}{cc}4& 0\\ 0& 9\end{array}\right]$

Thus, we see that the resultant matrix, $A+B$, is also a diagonal matrix of the order $2\times 2$.

Let’s check matrix multiplication with the same matrices. We multiply Matrix $A$ and Matrix $B$ and show that the resultant is also a diagonal matrix with the same order. Shown below:

$A\times B=\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ 0& 5\end{array}\right]$

$A\times B=\left[\begin{array}{cc}3\times 1+0\times 0& 3\times 0+0\times 5\\ 0\times 1+4\times 0& 0\times 0+4\times 5\end{array}\right]$

$A\times B=\left[\begin{array}{cc}3& 0\\ 0& 20\end{array}\right]$

Thus, we see that the resultant matrix, $A\times B$, is also a diagonal matrix of the order $2\times 2$.

To learn more about how we did matrix multiplication, please have a look at the article here.

Property 2:

The transpose of a diagonal matrix is the matrix itself.

If we have a matrix $A$, then we denote its transpose as $A^T$. Transposing a matrix means to flip its rows and columns. Let’s show that this property is true by calculating the transpose of Matrix $A$.

$A=\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]$

$A^T=\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]$

Interchanging the rows and columns produces the same matrix because of the entries besides the diagonal being $0$.

Property 3:

Under multiplication, diagonal matrices are commutative. 

If we have $2$ matrices, $A$ and $B$, this means $AB=BA$. Let’s show this property by using the two matrices from above.

$A\times B=\left[\begin{array}{cc}3\times 1+0\times 0& 3\times 0+0\times 5\\ 0\times 1+4\times 0& 0\times 0+4\times 5\end{array}\right]$

$A\times B=\left[\begin{array}{cc}3& 0\\ 0& 20\end{array}\right]$

Now,

$B\times A=\left[\begin{array}{cc}1\times 3+0\times 0& 1\times 0+0\times 4\\ 0\times 3+5\times 0& 0\times 0+5\times 4\end{array}\right]$

$B\times A=\left[\begin{array}{cc}3& 0\\ 0& 20\end{array}\right]$

Thus, we have seen that $AB=BA$.

Determinant of Diagonal Matrix

First, let’s look at the determinant of a $2\times 2$ matrix.

Consider Matrix $M$ shown below:

$M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

The determinant of this matrix is:

$det(M)=ad–bc$

One property of a diagonal matrix is that the determinant of a diagonal matrix is equal to the product of the elements in its principal diagonal.

Let’s see if it’s true by finding the determinant of the diagonal matrix shown below.

$N=\left(\begin{array}{cc}2& 0\\ 0& 8\end{array}\right)$

$det(N)=(2\times 8)-(0\times 0)=16$

This is in fact the product of the elements in its diagonal, $2\times 8=16$.

Example 1

For the matrices shown below, identify whether they are diagonal matrix or not.

$A=\left[\begin{array}{cc}-2& 0\\ 0& -7\end{array}\right]$

$B=\left[\begin{array}{cc}a& 0\\ 0& b\\ 0& d\end{array}\right]$

$C=\left[\begin{array}{ccc}3& 0& 0\\ 0& 0& 0\\ 0& 0& 11\end{array}\right]$

$D=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Solution

• Matrix A is a $2\times 2$ matrix with the elements being 0 other than the diagonal. So, this is a diagonal matrix.
• Matrix B is a $3\times 2$ matrix. It’s not square, so immediately we can say that it is not a diagonal matrix.
• Matrix C is a square matrix ($3\times 3$). Also all the elements besides the diagonal are $0$. So, it is a diagonal matrix. Moreover, an entry of the diagonal is also $0$, it doesn’t matter as long as all the entries except the diagonal are zeros.
• Matrix D is a special type of diagonal matrix. It is a zero matrix. Therefore, it is a diagonal matrix.

Example 2

Will multiplying Matrix A and Matrix B result in a diagonal matrix?

$A=\left[\begin{array}{cc}-9& 0\\ 0& 0\end{array}\right]$

$B=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$

Solution

Matrix A is a diagonal matrix but Matrix B is not. So, multiplying  Matrix A and B will not result in a diagonal matrix.

Example 3

Find the determinant of the matrix shown below:

$B=\left[\begin{array}{ccc}8& 0& 0\\ 0& 4& 0\\ 0& 0& -1\end{array}\right]$

Solution

Matrix B is a $3\times 3$ diagonal matrix. Recall that the product of all the entries of the diagonal of a diagonal matrix is its determinant. Thus, we simply multiply and find the answer:

$det(B)=8\times 4\times -1=–32$

Practice Questions

1. Identify which of the following matrices are diagonal matrices.
   $J=\left(\begin{array}{cc}0& 0\\ 0& -5\end{array}\right)$
   $K=\left(\begin{array}{cc}0& 2\\ 1& -1\end{array}\right)$
   $L=\left[\begin{array}{ccc}-3& 0& 0\\ 0& -5& 0\\ 0& 0& 3\end{array}\right]$
2. Calculate the determinant of the matrix shown below:
   $T=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 4\end{array}\right]$
3. Given
   $A=\left(\begin{array}{cc}2& 0\\ 0& -1\end{array}\right)$
   $B=\left(\begin{array}{cc}1& 0\\ 1& -2\end{array}\right)$

   Is $AB=BA$ ?

Answers

1. Matrix J is a square matrix. All the elements other than the principal diagonal are zeros. This is a diagonal matrix.
   Matrix K is a square matrix but not all the elements, except the diagonal, are zero. Thus, this is not a diagonal matrix. 
   Matrix L is a square matrix (3\times3). The elements other than the diagonal entries are zero. So, this is a diagonal matrix.
2. This is a diagonal matrix. We can find the determinant of this matrix by taking the product of all the 3 entries of the diagonal. Thus, the determinant is:
   $det(T)=-1\times -1\times 4=4$
3. If two matrices are diagonal, the multiplication of those two matrices are commutative. Looking at Matrix A and B, we can see that Matrix A is diagonal but Matrix B is not. Hence, their multiplication will not be commutative.

   Thus, $AB\ne BA$.

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