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primitive matrix (Definition)

A nonnegative square matrix $ A=(a_{ij})$ is said to be a primitive matrix if there exists $ k$ such that $ A^k\gg 0$, i.e., if there exists $ k$ such that for all $ i,j$, the $ (i,j)$ entry of $ A^k$ is positive.

A sufficient condition for a matrix to be a primitive matrix is for the matrix to be a nonnegative, irreducible matrix with a positive element on the main diagonal.



"primitive matrix" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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Cross-references: diagonal, positive element, irreducible matrix, sufficient, positive, matrix, square matrix
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This is version 9 of primitive matrix, born on 2002-12-22, modified 2007-10-26.
Object id is 3809, canonical name is PrimitiveMatrix.
Accessed 7769 times total.

Classification:
AMS MSC: 15A48 (Linear and multilinear algebra; matrix theory :: Positive matrices and their generalizations; cones of matrices)
 15A51 (Linear and multilinear algebra; matrix theory :: Stochastic matrices)
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