Identity matrix

Properties

When ${\displaystyle A}$ is an ${\displaystyle m\times n}$ matrix, it is a property of matrix multiplication that $${\displaystyle I_{m}A=A{I}_n=A.}$$ In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all ${\displaystyle n\times n}$ matrices, and as the identity element of the general linear group ${\displaystyle \mathrm{G}\mathrm{L}(n)}$, which consists of all invertible ${\displaystyle n\times n}$ matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is an involutory matrix, equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.

When ${\displaystyle n\times n}$ matrices are used to represent linear transformations from an ${\displaystyle n}$-dimensional vector space to itself, the identity matrix ${\displaystyle I_n}$ represents the identity function, for whatever basis was used in this representation.

The ${\displaystyle i}$th column of an identity matrix is the unit vector ${\displaystyle e_i}$, a vector whose ${\displaystyle i}$th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is ${\displaystyle n}$.

The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:

1. When multiplied by itself, the result is itself
2. All of its rows and columns are linearly independent.

The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.

The rank of an identity matrix ${\displaystyle I_n}$ equals the size ${\displaystyle n}$, i.e.: $${\displaystyle \mathrm{rank}(I_n)=n.}$$