An idempotent matrix is a square matrix that, when multiplied by itself, yields the same matrix. In mathematical terms, a matrix A is idempotent if A^2 = A . This property means that applying the matrix transformation twice is equivalent to applying it once. Idempotent matrices are significant in various fields, including linear algebra and statistics. They are often used in projection operations, where they project vectors onto a subspace without changing the outcome after the first application. This characteristic makes them useful in data analysis and machine learning for simplifying complex calculations.