# Diagonal Matrices

Definition: A square $n \times n$ matrix $D$ is a Diagonal Matrix if all entries off the main diagonal are zero, that is for all $i \neq j$, $a_{ij} = 0$. |

The following three matrices $A$, $B$, and $C$ are all diagonal matrices. Note that the entries of the main diagonal are not pertinent in the definition of a diagonal matrix, and so entries in the main diagonal can be zero.

(1)If we have two matrices $A$ and $D$ with $D$ being a diagonal matrix (assuming multiplication is defined), then calculating the product of these matrices is relatively easy. If we want to find the product $AD$, then we multiply the first column of $A$ by $d_1$, the second column of $A$ by $d_2$, …, and the $n^{\mathrm{th}}$ column of $A$ by $d_n$:

(2)If we want to find the product $DA$, we multiply the first row of $A$ by $d_1$, the second row of $A$ by $d_2$, …, and the $n^{\mathrm{th}}$ row of $A$ by $d_n$:

(3)We will now look at some examples of calculating matrix products involving diagonal matrices.

# Example 1

**Given the following matrices, determine the product $AB$.**

We note that $B$ is a diagonal matrix and to determine the product $AB$, we can multiply column $1$ of $A$ by $b_{11}$, column $2$ of $A$ by $b_{22}$, and column $3$ of $A$ by $b_{33}$:

(5)## Example 2

**Prove that if $A$ is a diagonal matrix, then $A = A^T$.**

**Proof:**The transpose of a matrix only interchanges entries that are not on the main diagonal. If $A$ is a diagonal matrix, then the only entries that are interchanged are zeroes and the main diagonal is untouched. Additionally, a diagonal matrix is square so $A^T$ is also square. Recall two matrices are equal if they are both the same size and all corresponding entries are diagonal, both of which were proved. $\blacksquare$