Definition:Diagonal Matrix

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix}$ be a square matrix of order $n$.

Then $\mathbf A$ is a diagonal matrix if and only if all elements of $\mathbf A$ are zero except for possibly its diagonal elements.


Thus $\mathbf A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ \end{bmatrix}$.


It follows by the definition of triangular matrix that a diagonal matrix is both an upper triangular matrix and a lower triangular matrix.


Also see

  • Results about diagonal matrices can be found here.


Sources