Define the Diagonal matrix.

A diagonal matrix is a square matrix where all non-diagonal elements are zero.

A diagonal matrix is a special type of square matrix where all non-diagonal elements are zero. This means that the only non-zero elements are on the diagonal from the top left to the bottom right. For example, the matrix below is a diagonal matrix:

$$
\begin{pmatrix}
3 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 2 \\
\end{pmatrix}
$$

Diagonal matrices are often used in linear algebra because they have some useful properties. For example, multiplying a diagonal matrix by a vector simply scales each element of the vector by the corresponding diagonal element. This makes diagonal matrices useful for representing scaling transformations.

Diagonal matrices are also easy to invert. To invert a diagonal matrix, we simply take the reciprocal of each non-zero diagonal element. For example, the inverse of the matrix above is:

$$
\begin{pmatrix}
1/3 & 0 & 0 \\
0 & 1/5 & 0 \\
0 & 0 & 1/2 \\
\end{pmatrix}
$$

Finally, diagonal matrices can be added and multiplied together in a straightforward way. When adding two diagonal matrices, we simply add the corresponding diagonal elements. When multiplying two diagonal matrices, we multiply the corresponding diagonal elements. For example:

$$
\begin{pmatrix}
3 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 2 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 4 \\
\end{pmatrix}
=
\begin{pmatrix}
3\times 1 & 0 & 0 \\
0 & 5\times 2 & 0 \\
0 & 0 & 2\times 4 \\
\end{pmatrix}
=
\begin{pmatrix}
3 & 0 & 0 \\
0 & 10 & 0 \\
0 & 0 & 8 \\
\end{pmatrix}
$$