Diagonal Matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero

A diagonal matrix can be constructed from a vector $\mathbf{a}=(a_1,\dots ,a_n{)}^t$:

$$\left(\begin{array}{cccc}{a}_1& 0& \cdots & 0\\ 0& a_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_n\end{array}\right)=\mathrm{diag}(a_1,a_2,\dots ,a_n)$$

Matrix Multiplication

$$\mathrm{diag}(a_1,\dots ,a_n)\mathrm{diag}(b_1,\dots ,b_n)=\mathrm{diag}(a_1{b}_1,\dots ,a_n{b}_n)$$

which implies,

$$\mathrm{diag}(a_1,a_2,\dots ,a_n)=\mathrm{diag}(a_1^k,a_2^k,\dots ,a_n^k)$$

Matrix Determinant

$$\mathrm{diag}(a_1,a_2,\dots ,a_n)=a_1{a}_2\cdots a_n$$