Identity matrix

Not to be confused with matrix of ones.

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.) Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix"[1] and the German word "Einheitsmatrix",[2] respectively.

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} ,\ \cdots ,\ 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}

When A is m×n, it is a property of matrix multiplication that

I_mA = AI_n = A. \,

In particular, the identity matrix serves as the unit of the ring of all n×n matrices, and as the identity element of the general linear group GL(n) consisting of all invertible n×n matrices. (The identity matrix itself is invertible, being its own inverse.)

Where n×n matrices are used to represent linear transformations from an n-dimensional vector space to itself, In represents the identity function, regardless of the basis.

The ith column of an identity matrix is the unit vector ei. It follows that the determinant of the identity matrix is 1 and the trace is n.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

I_n = \mathrm{diag}(1,1,...,1). \,

It can also be written using the Kronecker delta notation:

(I_n)_{ij} = \delta_{ij}. \,

The identity matrix also has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another.

The identity matrix of a given size is the only idempotent matrix of that size having full rank. That is, it is the only matrix such that (a) when multiplied by itself the result is itself, and (b) all of its rows, and all of its 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.[3]

See also

Notes

  1. ↑ Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
  2. ↑ "Identity Matrix" on MathWorld;
  3. ↑ Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of I2". The Mathematical Gazette 87, November 2003, 499-500.

External links

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