**What is Diagonal Matrix?**

In mathematics, the term diagonals matrix define as the **matrix** in which the off diagonals entries are zero and main diagonals entries are some else. There is no restriction for main diagonals entries. diagonals matrix is also called the **scaling matrix** because when we multiply any matrix with diagonals matrix it change the scale of that matrix here the scale meaning size of the matrix. The** identity matrix** is one of the examples of the diagonals matrix. because in the identity matrix the main diagonals are 1 and off diagonals are zero.

** Diagonal Matrix Definition?**

Diagonals matrix must be a **square matrix**. Since the square matrix is a symmetric matrix so we also called this matrix the symmetric diagonals matrix.

**Examples of diagonals matrix**

**Example no 1**:

\begin{vmatrix}6&0\\0&4\end{vmatrix}
It is a 2 by 2 diagonals matrix.

**Example no 2**:

\;\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}
It is 3 by 3 diagonals matrix.

**Example no 3: **

\begin{bmatrix}7&0&0&0\\0&5&0&0\\0&0&4&0\\0&0&0&9\end{bmatrix}
it is a 4 by 4 diagonals matrix.

**Properties of Diagonals matrix**

- When we add two diagonals matrices then the resultant matrix also diagonals matrix. \;\begin{bmatrix}1&0&0\\0&4&0\\0&0&3\end{bmatrix}\;\;+\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&9&0\\0&0&5\end{bmatrix}
- If any matrix is triangular and normal then then it is diagonal matrix.
- both the upper and lower triangular matrix are diagonals matrix.
- Symmetric matrix also diagonals matrix.
- When we multiply any matric with diagonals matric then the matrix which we obtain is diagonals matrix \;\begin{bmatrix}2&0&0\\0&4&0\\0&0&5\end{bmatrix}\;\;\times\;\;\begin{bmatrix}4&0&0\\0&3&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&12&0\\0&0&10\end{bmatrix}
- Zero matrix is diagonals matrix. \;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\; it is a 3 by 3 zero matrix and it hold the property of diagonal matrix.
- Identity matrix also diagonals matrix \;\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} this is the 3 by 3 identity matrix since it is hold the property of diagonals matrix
- The matrix of order 1 by 1 is diagonals matrix.
- Transpose of any diagonals matrix is same as matrix.
- Inverse of diagonals matrix is resiprocal of main diagonals entries .
- in under multiplication the diagonals matrix are commutative as AB = BA

**Note Diagonal Matrix**

- since 1 by 1 order matrix is also diagonal matrix.
- there is no restrication for diagonal entries.