n×nIdentitätsmatrix

The n×n identity matrix, often denoted as Iₙ or simply I when the size is clear from context, is a square matrix of size n×n with ones on the main diagonal and zeros elsewhere. It serves as the multiplicative identity in the ring of n×n matrices, meaning that for any n×n matrix A, the equation IₙA = AIₙ = A holds. This property makes the identity matrix a fundamental concept in linear algebra and matrix theory.

The identity matrix can be thought of as the matrix analogue of the number 1 in scalar multiplication. Just as multiplying any number by 1 leaves the number unchanged, multiplying any matrix by the identity matrix of the appropriate size leaves the original matrix unchanged. This property is crucial in various mathematical operations and proofs involving matrices.

The identity matrix has several important properties. Firstly, it is idempotent, meaning that Iₙ multiplied by itself any number of times results in the identity matrix itself (Iₙᵏ = Iₙ for any positive integer k). Secondly, the determinant of the identity matrix is always 1, and its trace (the sum of the elements on the main diagonal) is n. Additionally, the identity matrix is its own inverse, as Iₙ⁻¹ = Iₙ.

In practical applications, the identity matrix is used in solving systems of linear equations, in the definition of matrix inverses, and in various transformations in computer graphics and engineering. Its simplicity and fundamental properties make it an essential tool in both theoretical and applied mathematics.