Identity matrix

An identity matrix is a square matrix with ones on its main diagonal and zeros elsewhere. Multiplying a compatible matrix by it leaves that matrix unchanged.

Matrix multiplication: An operation that combines matrices by taking row-by-column products to produce another matrix. The identity matrix leaves a compatible matrix unchanged under this operation.
Square matrix: A matrix with the same number of rows and columns. Identity matrices are square because their diagonal must span equally many rows and columns.
Gaussian elimination: A method for solving linear systems by reducing an augmented matrix through elementary row operations. Row operations can transform a coefficient matrix toward the identity to expose its inverse or solve a system.
Scalar multiplication: Multiplication of a vector or matrix by a single number. Scaling by one leaves entries unchanged, but the identity matrix acts through matrix multiplication.
Identity element: An element that leaves other elements unchanged when combined with them by an operation. The identity matrix is the identity element for square matrices of a fixed size under multiplication.
Main diagonal: The entries of a matrix whose row and column indices are equal. The identity matrix places all its ones on this diagonal.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. The identity matrix represents the transformation that leaves every vector unchanged.
Hadamard product: Entrywise multiplication of two matrices of the same dimensions. Under entrywise multiplication, the identity matrix does not preserve arbitrary matrices.
Inverse matrix: A matrix that multiplies with a given square matrix to produce the identity matrix. The identity matrix is the product that defines a matrix inverse.
Zero matrix: A matrix whose entries are all zero. Unlike the zero matrix, the identity matrix has diagonal entries that preserve multiplication.

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