What is an Irreducible Matrix?

An irreducible matrix is a square matrix that cannot be decomposed into the product of two smaller matrices, i.e., it cannot be written as the product of two matrices of smaller dimensions. In other words, a matrix is irreducible if it cannot be diagonalized by similarity transformation.

For example, a 2x2 identity matrix is irreducible because it cannot be decomposed into the product of two smaller matrices. A 3x3 matrix with no zero elements on its main diagonal is also irreducible because it cannot be diagonalized by similarity transformation.

In linear algebra, irreducible matrices are important in many applications, such as eigenvalue decomposition, linear transformations, and Markov chains.

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