Ominaispolynomi

An Ominaispolynomi, known in English as a minimal polynomial, is a fundamental concept in linear algebra and polynomial algebra associated with matrices and linear transformations. It is defined for a square matrix \(A\) over a field \(F\) as the monic polynomial \(m_A(x)\) of least degree with coefficients in \(F\) such that \(m_A(A) = 0\), where \(0\) denotes the zero matrix.

The minimal polynomial has several important properties. It divides the characteristic polynomial of the matrix, which is obtained from the determinant \(\det(xI - A)\). The roots of the minimal polynomial are precisely the eigenvalues of \(A\), and each eigenvalue appears in the minimal polynomial with an exponent equal to the size of the largest Jordan block associated with that eigenvalue.

The minimal polynomial provides insight into the structure of a matrix, particularly concerning diagonalizability. A matrix is diagonalizable over \(F\) if and only if its minimal polynomial factors into distinct linear factors over \(F\), meaning it contains no repeated roots.

In the context of algebraic structures, the minimal polynomial can also be associated with algebraic elements over a field, where it is the monic polynomial of least degree over \(F\) that has the element as a root. In this setting, it serves as an essential tool for understanding algebraic extensions and polynomial factorizations.

Overall, the minimal polynomial is a key concept connecting eigenvalues, matrix structure, and algebraic properties, playing a central role in linear algebra, polynomial theory, and algebraic field extensions.