World Library  
Flag as Inappropriate
Email this Article

Identity matrix

Article Id: WHEBN0000059718
Reproduction Date:

Title: Identity matrix  
Author: World Heritage Encyclopedia
Language: English
Subject: Determinant, Lorentz transformation, Bargmann–Wigner equations, Relativistic quantum mechanics, Matrix difference equation
Collection: 1 (Number), Matrices, Sparse Matrices
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Identity matrix

In linear algebra, the identity matrix or 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

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.