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The Jordan Canonical Form (JCF) is a powerful matrix representation that decomposes any square matrix into a block diagonal form consisting of Jordan blocks, where each block corresponds to an eigenvalue and contains 1s on the superdiagonal. This form is essential for understanding non-diagonalizable matrices, as it reveals the algebraic and geometric multiplicities of eigenvalues through the structure of Jordan blocks. The JCF is particularly valuable for solving systems of differential equations, computing matrix functions, and analyzing the structure of linear operators.