Identity matrix

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In mathematics, the identity matrix or identity matrix is a square matrix whose elements are one on the main diagonal and zero everywhere else. The identity matrix is ​​the neutral element in the ring of square matrices with regard to the matrix multiplication . It is symmetrical , self-inverse , idempotent and has maximum rank . The identity matrix is ​​the representation matrix of the identity mapping of a finite-dimensional vector space . It is used, among other things, in the definition of the characteristic polynomial of a matrix, orthogonal and unitary matrices, as well as in a number of geometric maps.


If a ring with zero element and one element , then the identity matrix is the square matrix


An identity matrix is ​​therefore a diagonal matrix in which all elements on the main diagonal are the same . In addition to (of identity ), (of unity ) is also used as a spelling . If the dimension is evident from the context, the index is often omitted and only written or .


If the field of real numbers and denotes and the numbers zero and one , then examples of unit matrices are:



The elements of an identity matrix can be expressed using the Kronecker delta

specify. The identity matrix of size can be just as simple

be noted. The rows and columns of the identity matrix are the canonical unit vectors , and one writes accordingly


when the unit vectors are column vectors.


The following applies to every matrix


Accordingly, the product of any matrix with the identity matrix results in the same matrix again. The set of square matrices, together with the matrix addition and the matrix multiplication, form a (non-commutative) ring . The identity matrix is ​​then the unitary element in this matrix ring , i.e. the neutral element with regard to the matrix multiplication.


The identity matrix is symmetric , that is for their transpose applies


and self-inverse , that is, for its inverse , also applies



The following applies to the determinant of the identity matrix


which is one of the three defining properties of a determinant. The following applies to the trace of the identity matrix


If it is the ring is , , or , is accordingly obtained . The characteristic polynomial of the identity matrix results as


The only eigenvalue is therefore with multiplicity . In fact applies to all of the module . If a commutative ring , then the rank of the identity matrix is ​​through



The identity matrix is idempotent , that is


and it is the only full ranked matrix with this property. The following applies to the matrix exponential of a real or complex unit matrix


where is Euler's number .


Linear Algebra

The set of regular matrices of size, together with the matrix multiplication, form the general linear group . Then applies to all matrices of this group and their inverses


The center of this group is precisely the multiples (not equal to zero) of the identity matrix. According to the definition, the following applies to an orthogonal matrix

and accordingly for a unitary matrix


These matrices each form subgroups of the corresponding general linear group. The zeroth power of a square matrix is called

set. The identity matrix is ​​used further in defining the characteristic polynomial

a square matrix is ​​used. The identity matrix is ​​the representation matrix of the identity mapping of a finite-dimensional vector space .


In analytical geometry , unit matrices are used, among other things, to define the following mapping matrices:


In the numerical software package MATLAB , the unit matrix of the size is generated by the function . In Mathematica , the identity matrix is ​​obtained by . eye(n)IdentityMatrix[n]

See also

  • Ones matrix , a matrix made up of only ones
  • Zero matrix , a matrix that consists of only zeros
  • Standard matrix, a matrix that consists of exactly one one and otherwise only zeros
  • Permutation matrix , a matrix that is created from a unit matrix by swapping rows or columns
  • Elementary matrix, a matrix that differs from an identity matrix only in one position or by exchanging rows


Web links

Individual evidence

  1. Christoph W. Überhuber, Stefan Katzenbeisser, Dirk Praetorius: MATLAB 7: An Introduction . Springer, 2007, p. 18 .