Identity Matrix¶

Definition¶

The identity matrix $\boldsymbol{I}_n$ is the $n\times n$ matrix with $1$ on the main diagonal and $0$ everywhere else.

Interpretation¶

The identity matrix acts like the number $1$ for matrix multiplication: multiplying by it changes nothing. Its columns are the standard basis vectors $\boldsymbol{e}_1,\ldots,\boldsymbol{e}_n$ of $\mathbb{R}^n$.

The left part shows $\boldsymbol{I}_3$ with its ones on the main diagonal shaded. The right part shows the map $\boldsymbol{v}\mapsto\boldsymbol{I}\boldsymbol{v}$ in the plane, given by $\boldsymbol{I}_2$: the square and the vector $\boldsymbol{v}$ stay where they are, since $\boldsymbol{I}\boldsymbol{v}=\boldsymbol{v}$$\mathbf{v}$. As a linear map, $\boldsymbol{I}$ is the identity function.

The reason is visible in the row-times-column rule: row $i$ of $\boldsymbol{I}$ has a single $1$, in position $i$, so multiplying it with a column picks out the $i$th entry of that column and nothing else. In the language of algebraic structures, $\boldsymbol{I}_n$ is the identity element for multiplication of $n\times n$ matrices. The subscript gives the size and is dropped when the size is clear from the product; for a non-square $\boldsymbol{A}$, the identity on the left and the one on the right have different sizes.

The identity matrix is also the target of Gauss-Jordan elimination: when the coefficient part of an augmented matrix has become $\boldsymbol{I}$, the system has a unique solution, which stands in the last column. Elementary matrices are obtained from $\boldsymbol{I}$ by one row operation. The identity matrix returns later in the definition of the inverse matrix, $\boldsymbol{A}^{-1}\boldsymbol{A}=\boldsymbol{I}$$\mathbf{I}$.

Key Results¶

• For every $m\times n$ matrix $\boldsymbol{A}$, $\boldsymbol{I}_m\boldsymbol{A}=\boldsymbol{A}$ and $\boldsymbol{A}\boldsymbol{I}_n=\boldsymbol{A}$$\mathbf{A}$.
• The identity matrix is diagonal, symmetric, triangular, idempotent, and a permutation matrix.

Examples¶

• $\boldsymbol{I}_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$.
• $\boldsymbol{I}_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$.

Prerequisites¶

• Square Matrix

Related Concepts¶

• Identity Map

• Matrix Diagonals

• Diagonal Matrix

Sources¶

• Math II: Linear Algebra · Week 3
• Math V: Advanced Linear Algebra · Week 1 (revisited)

Aliases: unit matrix Keywords: identity operator Symbols: real numbers

Last updated: 20 October 2026 03:02