LinArg assignment 5

exercise

1. Interpolation

$P(x)=-\frac{1}{12}{x}^3+\frac{3}{4}{x}^2-\frac{2}{3}x+1$

2. Strictly diagonally dominant matrices

Assume $A\mathbf{x}=0$ for some nonzero $\mathbf{x}\in {\mathbb{R}}^m$ Pick the $k$-th row, so that $|x_k|$ is the biggest among $|x_1|,\dots ,|x_m|$ $0={\sum }_{j=1}^m{a}_{kj}{x}_j$ $⟹|a_{kk}||x_k|=\left|{\sum }_{j=1,j\ne k}^m|a_{kj}||x_j|\right|\le {\sum }_{j=1,j\ne k}^m|a_{kj}||x_j|\le |x_k|{\sum }_{j=1,j\ne k}^m|a_{kj}|$ If $|x_k|\ne 0$, then we get $|a_{kk}|\le {\sum }_{j=1,j\ne k}^m|a_{kj}|$ which contradicts strict diagonal dominance. Hence no nonzero $\mathbf{x}$ satisfies $A\mathbf{x}=0$. So $A$ is linear independent and thus invertible.