Diagonal matrix
A diagonal matrix is a matrix with non-zero values only on the diagonal. For example
\[\begin{pmatrix} \gc{1} & \kc{0} & \kc{0} \\ \kc{0} & \gc{2} & \kc{0} \\ \kc{0} & \kc{0} & \gc{3} \\ \end{pmatrix} \p\]The term “diagonal matrix” usually refers to a square matrix, but it can also be used to refer to a non-suqare matrix. The matrix will then have some columns or rows that consist of only zeros. For example:
\[\begin{pmatrix} \gc{1} & \kc{0} & \kc{0} & \kc{0} \\ \kc{0} & \gc{2} & \kc{0} & \kc{0} \\ \kc{0} & \kc{0} & \gc{3} & \kc{0} \\ \end{pmatrix} \text{ or} \begin{pmatrix} \gc{1} & \kc{0} & \kc{0} \\ \kc{0} & \gc{2} & \kc{0} \\ \kc{0} & \kc{0} & \gc{3} \\ \kc{0} & \kc{0} & \kc{0} \\ \end{pmatrix} \p\]Square diagonal matrices are a useful concept for many reason. This is primarily because their behavior is simple to understand. For example
- Multiplying a square diagonal matrix $\gc{\D}$ by a vector $\x$ is equivalent to multiplys each element $x_i$ of $\x$ by the scalar $\gc{D}_{ii}$
- Geometrically, this corresponds to the operation of stretching along the $i$-th axis by a factor of $\gc{D}_{ii}$.
- The eigenvalues of a geometric matrix are equal to the values on the diagonal.
$\text{diag}(\cdot)$
The operator $\text{diag}(\cdot)$ is used to construct a diagonal matrix, or to read the diagonal from a matrix, depending on whether the argument is a vector or a matrix.
If the argument is a matrix $\bc{\A}$ (diagonal or otherwise), then $\text{diag}(\bc{\A})$ refers to the vector $\bc{\a}$ for which the $i$-th element is the element $\bc{A}_{ii}$ of the matrix. That is, the operator slices the diagonal out of the matrix and returns the result as a vector
$$[\text{diag}(\bc{\A})]_i = \bc{A}_{ii} \p$$
If the argument is a vector $\bc{\a}$, say of dimension $n$ then $\text{diag}(\bc{\a})$ refers to the diagonal $n \times n$ matrix with the elements of $\bc{\a}$ along the diagonal.
$$[\text{diag}(\bc{\a})]_{ii} = \bc{a}_{i} \p$$