The Smith normal form of a matrix

V diplomskem delu je predstavljena Smithova normalna oblika kvadratne matrike z elementi iz kolobarja polinomov. Za podano kvadratno matriko izračunamo njeno Smithovo normalno obliko z uporabo elementarnih vrstičnih in stolpičnih transformacij. Za Smithovo normalno obliko matrike je značilno, da je diagonalna matrika s toliko neničelnimi elementi na diagonali, kolikšen je rang matrike, pri tem so diagonalni elementi unitarni polinomi ter prvi diagonalni element deli drugega, drugi deli tretjega in tako naprej do zadnjega neničelnega elementa na diagonali. Smithova normalna oblika matrike nam lahko služi kot orodje za ugotavljanje podobnosti matrik z elementi iz polja.In this work we present the Smith normal form of a square matrix with entries from the ring of polynomials. Given a square matrix its Smith normal form can be calculated using elementary row and column transformations. The Smith normal form of the matrix with rank r is a diagonal matrix with r non-zero elements on the diagonal, diagonal elements are unitary polynomials and the first diagonal element divides the second one, the second one divides the third one and so on until the last non-zero element on diagonal. Smith normal form of a matrix can be used as a tool for checking similarity of matrices with entries from a field