Diagonal Matrices

A Diagonal Matrix is a special kind of square matrix in which all the elements outside the main diagonal are zero. Only the elements on the main diagonal (from the top-left corner to the bottom-right corner) can be non-zero or zero.

Definition:

A square matrix $A=[a_{ij}]$ of order n x n is called a Diagonal Matrix if:

$a_{ij}=\{\begin{array}{ll}{a}_{ii},& \text{if }i=j\\ 0,& \text{if }i\ne j\end{array}$

In simple terms,

$A=\left[\begin{array}{ccccc}{a}_{11}& 0& 0& \cdots & 0\\ 0& a_{22}& 0& \cdots & 0\\ 0& 0& a_{33}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & a_{nn}\end{array}\right]$

Example of a Diagonal Matrix:

$\left[\begin{array}{ccc}5& 0& 0\\ 0& -3& 0\\ 0& 0& 2\end{array}\right]$

1.0What is a Diagonal Matrix?

A diagonal matrix is a type of square matrix where all elements outside the main diagonal are zero. In simpler terms, only the elements on the main diagonal (from top-left to bottom-right) can be non-zero, and everything else is zero.

2.0Diagonal Matrix Formula

Let $[a_{ij}]$be an n x n matrix. Then A is a diagonal matrix if:

$a_{ij}=0\text{ for all }i\ne j$

So, a general diagonal matrix of order n looks like:

$D=\left[\begin{array}{ccccc}{d}_1& 0& 0& \cdots & 0\\ 0& d_2& 0& \cdots & 0\\ 0& 0& d_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & d_n\end{array}\right]$

3.0Diagonal Matrix 2 x 2 Example

A diagonal matrix of order 2 x 2:

$D=\left[\begin{array}{cc}5& 0\\ 0& 3\end{array}\right]$

Here, all non-diagonal elements are 0.

You can also have zeros on the diagonal:

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

4.0Diagonal Matrix 3x3 Example

$D=\left[\begin{array}{ccc}1& 0& 0\\ 0& -4& 0\\ 0& 0& 6\end{array}\right]$

Clearly, the only non-zero elements are on the main diagonal.

5.0 Properties of a Diagonal Matrix

• All off-diagonal elements are 0.
• A scalar matrix is a special diagonal matrix where all diagonal elements are equal.
• The transpose of a diagonal matrix is itself.
• The product of two diagonal matrices is also a diagonal matrix.
• The inverse of a diagonal matrix (if all diagonal elements ≠ 0) is also a diagonal matrix.

6.0Diagonal Matrix Determinant

The determinant of a diagonal matrix is the product of its diagonal elements.

$\text{If }D=\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right],\text{ then }\det (D)=d_1\cdot d_2\cdot d_3$

Example:

$D=\left[\begin{array}{ccc}2& 0& 0\\ 0& 5& 0\\ 0& 0& 4\end{array}\right]⟹\det (D)=2\cdot 5\cdot 4=40$

7.0Diagonal Matrix Examples

Here are more examples for clarity:

Example 1 (2x2):

$A=\left[\begin{array}{cc}9& 0\\ 0& -1\end{array}\right]⟹\text{Diagonal matrix of order 2}$

Example 2 (3x3):

$B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 5\end{array}\right]⟹\text{Diagonal matrix with one non-zero entry}$

8.0Solved Examples on Diagonal Matrices

Example 1: Determine whether the following matrix is a diagonal matrix:

$A=\left[\begin{array}{ccc}5& 0& 0\\ 0& -3& 0\\ 0& 0& 7\end{array}\right]$

Solution:

A matrix is diagonal if all off-diagonal elements are zero and diagonal elements can be any number (including zero).

Here, all off-diagonal elements are 0, and diagonal elements are 5, -3, and 7.

Answer: Yes, A is a diagonal matrix.

Example 2: Let . $A=\left[\begin{array}{cc}2& 0\\ 0& 5\end{array}\right],B=\left[\begin{array}{cc}3& 0\\ 0& -1\end{array}\right]$Find A + B.

Solution:

$A+B=\left[\begin{array}{cc}2+3& 0+0\\ 0+0& 5+(-1)\end{array}\right]=\left[\begin{array}{cc}5& 0\\ 0& 4\end{array}\right]$

Answer:

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

Example 3: Let $A=\left[\begin{array}{cc}4& 0\\ 0& 3\end{array}\right],B=\left[\begin{array}{cc}2& 0\\ 0& 5\end{array}\right].$ Find A x B.

Solution:

Multiplying diagonal matrices is easy:

$AB=\left[\begin{array}{cc}4\times 2& 0\\ 0& 3\times 5\end{array}\right]=\left[\begin{array}{cc}8& 0\\ 0& 15\end{array}\right]$

Answer:

$AB=\left[\begin{array}{cc}8& 0\\ 0& 15\end{array}\right]$

Example 4: Find the inverse of $A=\left[\begin{array}{ccc}2& 0& 0\\ 0& -3& 0\\ 0& 0& 5\end{array}\right]$if it exists.

Solution:

A diagonal matrix is invertible if none of its diagonal elements are zero.

The inverse of a diagonal matrix is another diagonal matrix with reciprocals of the original diagonal elements.

So,

$A^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& -\frac{1}{3}& 0\\ 0& 0& \frac{1}{5}\end{array}\right]$

Answer:

$A^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& -\frac{1}{3}& 0\\ 0& 0& \frac{1}{5}\end{array}\right]$

Example 5: Find A^3, where $A=\left[\begin{array}{cc}2& 0\\ 0& -3\end{array}\right]$

Solution:

For a diagonal matrix, raising to a power means raising each diagonal element to that power.

So,

$A^3=\left[\begin{array}{cc}{2}^3& 0\\ 0& (-3{)}^3\end{array}\right]=\left[\begin{array}{cc}8& 0\\ 0& -27\end{array}\right]$

Answer:

$A^3=\left[\begin{array}{cc}8& 0\\ 0& -27\end{array}\right]$

9.0Practice Questions on Diagonal Matrices

1. Identify whether the following matrix is diagonal:

$\left[\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$

2. Find the determinant of:

$\left[\begin{array}{cc}3& 0\\ 0& -5\end{array}\right]$

3. Is the matrix $\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right]$ a scalar matrix?

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