thumb | right | Example of a 3x3 identity matrix
In linear algebra, the identity matrix of size <math>n</math> is the <math>n\times n</math> square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
Terminology and notation
The identity matrix is often denoted by <math>I_n</math>, or simply by <math>I</math> if the size is immaterial or can be trivially determined by the context.
<math display="block">
I_1 = \begin{bmatrix} 1 \end{bmatrix}
,\
I_2 = \begin{bmatrix}
1 & 0 \\
0 & 1 \end{bmatrix}
,\
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}
,\ \dots ,\
I_n = \begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \end{bmatrix}.
</math>
The term unit matrix has also been widely used, but the term identity matrix is now standard. The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all <math>n\times n</math> matrices.
In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, <math>\mathbf{1}</math>, or called "id" (short for identity). Less frequently, some mathematics books use <math>U</math> or <math>E</math> to represent the identity matrix, standing for "unit matrix"
In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as
<math display=block> I_n = \operatorname{diag}(1, 1, \dots, 1).</math>
The identity matrix can also be written using the Kronecker delta notation:
The rank of an identity matrix <math>I_n</math> equals the size <math>n</math>, i.e.:
<math display=block>\operatorname{rank}(I_n) = n .</math>
See also
- Binary matrix (zero-one matrix)
- Elementary matrix
- Exchange matrix
- Matrix of ones
- Pauli matrices (the identity matrix is the zeroth Pauli matrix)
- Householder transformation (the Householder matrix is built through the identity matrix)
- Square root of a 2 by 2 identity matrix
- Unitary matrix
- Zero matrix