Diagonal Matrix

A diagonal matrix is a Matrix ${\hbox{\sf A}}$ of the form

$$a_{ij} = c_i\delta_{ij},$$ (1)

where $\delta$ is the Kronecker Delta, $c_i$ are constants, and there is no summation over indices. The general diagonal matrix is therefore Square and of the form

$$\left[{\matrix{ c_1 & 0 & \cdots & 0\cr 0 & c_2 & \cdots &... ...\vdots & \ddots & \vdots\cr 0 & 0 & \cdots & c_n\cr}}\right].$$ (2)

Given a Matrix equation of the form

$$\left[{\matrix{a_{11} & \cdots & a_{1n}\cr \vdots & \ddots &... ...ots & \ddots & \vdots\cr a_{n1} & \cdots & a_{nn}\cr}}\right],$$ (3)

multiply through to obtain

$$\left[{\matrix{a_{11}\lambda_1 & \cdots & a_{1n}\lambda_n\cr... ...ots\cr a_{n1}\lambda_n & \cdots & a_{nn}\lambda_n\cr}}\right].$$ (4)

Since in general, $\lambda_i \not= \lambda_j$ for $i\not=j$, this can be true only if off-diagonal components vanish. Therefore, A must be diagonal.

Given a diagonal matrix ${\hbox{\sf T}}$,

$${\hbox{\sf T}}^n=\left[{\matrix{ t_1 & 0 & \cdots & 0\cr 0... ...ts & \ddots & \vdots\cr 0 & 0 & \cdots & {t_k}^n\cr}}\right].$$ (5)

See also Matrix, Triangular Matrix, Tridiagonal Matrix

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 181-184 and 217-229, 1985.