Generic matrix

The term generic matrix is used in various meanings in the mathematical branch of linear algebra .

In representation theory, matrices are called matrices in which (for all ) the sub-matrices formed from the last rows and first columns have determinants different from zero , see Bruhat decomposition # Generic matrices . ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$

Occasionally also the matrix

${\displaystyle \left({\begin{array}{cccc}x_{11}&x_{12}&\ldots &x_{1n}\\x_{21}&x_{22}&\ldots &x_{2n}\\\ldots &\ldots &&\ldots \\x_{n1}&x_{n2}&\ldots &x_{nn}\end{array}}\right)\in Mat(n\times n,\mathbb {Z} \left[x_{ij}\right])}$

called generic matrix in algebraically independent variables . This generic matrix is ​​used, for example, in proving the Cayley-Hamilton Theorem . ${\displaystyle x_{ij},1\leq i,j\leq n}$

There are other uses of the term in the mathematical specialist literature that are incompatible with the above.

See also

• Generic property