How do i work out if this matrix is diagonally dominted?

Question

I am having trouble answering these two questions: (a) Is matrix A diagonally dominant? (b) Find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices.

Answer 1

A matrix is diagonally dominant if in every row, the absolute value of the main diagonal is greater than the sum of absolute values of the rest of the row. In other words, $$ |a_{ii}| > \sum_{k\ne i} |a_{ik}|. $$

Some examples are $$ \begin{pmatrix} 2 & 1 \\ 1 & -2\end{pmatrix}, \begin{pmatrix} 5 & 3 & 1 \\ 4 & -5 & 0 \\ 1 & 1 & -3\end{pmatrix} $$

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Spectral radius is the largest size of the eigenvalue of that matrix.

For example, the spectral radius of $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -3\end{pmatrix} $$ is $\max \{|1|, |-5|, |-3|\} = 5$.

To find the spectral radius, one way is to find all the eigenvalues, compute their sizes and take the largest one.