Matrices with MathSpeak

Numeric Matrix

\[ \left[ {\begin{array}{*{20}{r}}5&4&2&1\\[2pt]0&1&-1&-1\\[2pt]-1&-1&3&0\\[2pt]1&1&-1&2\end{array}} \right] \]

A Cross Product Formula

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix} \]

Jordan Normal Form

\[ J_i = \begin{bmatrix} \lambda_i & 1 & \; & \; \\ \; & \lambda_i & \ddots & \; \\ \; & \; & \ddots & 1 \\ \; & \; & \; & \lambda_i \end{bmatrix} \]

Jordan Decomposition

\[ f\left({\begin{bmatrix}\lambda &1&0&\ldots &0\\0&\lambda &1&\vdots &\vdots \\0&0&\ddots &\ddots &\vdots \\\vdots &\ldots &\ddots &\lambda &1\\0&\ldots &\ldots &0&\lambda \end{bmatrix}}\right)={\begin{bmatrix}{\frac {f(\lambda )}{0!}}&{\frac {f'(\lambda )}{1!}}&{\frac {f''(\lambda )}{2!}}&\ldots &{\frac {f^{(n)}(\lambda )}{n!}}\\0&{\frac {f(\lambda )}{0!}}&{\frac {f'(\lambda )}{1!}}&\vdots &{\frac {f^{(n-1)}(\lambda )}{(n-1)!}}\\0&0&\ddots &\ddots &\vdots \\\vdots &\ldots &\ddots &{\frac {f(\lambda )}{0!}}&{\frac {f'(\lambda )}{1!}}\\0&\ldots &\ldots &0&{\frac {f(\lambda )}{0!}}\end{bmatrix}}. \]