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Definition 01 · Matrices

Identity matrix

The neutral element of matrix multiplication: it leaves every compatible vector and matrix unchanged.

In
Definition

The identity matrix of order n is the n × n square matrix with 1 in every position on the main diagonal and 0 everywhere else.

Form and notation

The symbol In records the order of the matrix. Select an order to see how only the diagonal entries change to one.

Choose the order n

I3 = 100 010 001
(In)ij = 1 when i = j, and 0 otherwise

Key properties

Its role mirrors the number 1 in ordinary multiplication. The dimensions must always be compatible with the matrix or vector beside it.

ImA = A = AIn

Multiplicative identity. For an m × n matrix A, multiplication by the correctly sized identity changes nothing.

In−1 = In

Its own inverse. Applying the identity again still does nothing, so it undoes itself.

det(In) = 1

Unit determinant. The identity preserves every area, volume, and higher-dimensional scale factor.

Inv = v

Every vector is fixed. Each nonzero vector is an eigenvector of In with eigenvalue 1.

Geometric meaning

Matrices can describe transformations such as rotation, reflection, scaling, and shearing. The identity matrix describes the transformation that does none of them.

Everything stays put.

Multiplying a vector by In preserves every coordinate. Its direction, length, and position remain exactly the same.

A common pitfall

The identity matrix is not the same as a matrix filled with ones. Only its main diagonal contains ones; every off-diagonal entry is zero.

Dimension matters. There is a different identity matrix for every positive order. For example, use I2 with two-dimensional vectors and I3 with three-dimensional vectors.
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