A standard matrix , standard unit matrix or matrix unit is in mathematics a matrix in which one entry is one and all other entries are zero. Every standard matrix can be represented as a dyadic product of canonical unit vectors . The set of standard matrices forms the standard basis for the matrix space . Among other things, they are used to define elementary matrices that are used in the Gaussian elimination process .
definition
If a ring with zero element and one element , then the standard matrix is the matrix with the entries
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![E _ {{ij}} = (e _ {{kl}}) \ in R ^ {{m \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b416d374d37f12ae8704ca83cfc1968fdff6f8bb)
![e _ {{kl}} = {\ begin {cases} 1 & {\ text {for}} i = k {\ text {and}} j = l \\ 0 & {\ text {otherwise}} \ end {cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b9eb9294ab90a6bd2ba2238e6ea8b04ea8c7ef)
for and . In the standard matrix, the entry at this point is therefore equal to one and all other entries are equal to zero. A standard matrix is also known as a standard identity matrix or matrix unit and is occasionally notated with instead of .
![k = 1, \ ldots, m](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a3c5f7ce97004561fe563631144135783e8164)
![l = 1, \ ldots, n](https://wikimedia.org/api/rest_v1/media/math/render/svg/f741b515dd45d8312937a4e595f8a560cd5b8ceb)
![E _ {{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcb625c128efafee881204113cd9e7a8a293a0)
![(i, j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712)
![e_ {ij}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6a2274e22dc1d2778c28f3ce5b946d90ba2756)
![E _ {{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbcb625c128efafee881204113cd9e7a8a293a0)
Examples
If the field of the real numbers and denotes and the numbers zero and one , then examples of standard matrices are :
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![3 times 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e)
![E _ {{11}} = {\ begin {pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \ end {pmatrix}}, E _ {{12}} = {\ begin {pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \ end { pmatrix}}, E _ {{23}} = {\ begin {pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18d9bc89f4543b82639bf8e096cf9f3036b4902b)
properties
Representations
Every standard matrix can be represented as the dyadic product of the two canonical unit vectors and , that is
![e_ {i} \ in R ^ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd5c93e689cd632a3126ec37f09ce5df9d2a1a8)
![e_ {j} \ in R ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c809d9bf8075c4286a4ba2f8940c2bd04932c6c)
-
,
where the transposed vector is to. With the help of the Kronecker delta , a standard matrix can also be passed through
![e ^ {T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b37958eca1d75315b998189f3b26bf163d5c52d)
![e](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467)
![E _ {{ij}} = (\ delta _ {{ik}} \, \ delta _ {{jl}}) _ {{k = 1, \ ldots, m \ atop l = 1, \ ldots, n}} = (\ delta _ {{(i, j), (k, l)}}) _ {{k = 1, \ ldots, m \ atop l = 1, \ ldots, n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df729244e304a60272d6f98b4c82105602b84612)
note.
symmetry
The following applies to the transpose of a standard matrix
![E _ {{ij}} \ in R ^ {{{m \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe369f9c64caf6b8e3ef18c956592f7872c9f4)
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.
This means that only the standard matrices are symmetrical .
product
For the product of two standard matrices and applies
![E _ {{ij}} \ in R ^ {{{m \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe369f9c64caf6b8e3ef18c956592f7872c9f4)
![E _ {{kl}} \ in R ^ {{n \ times p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d7ba6aeb9a06222293f921f618747e363f46c0)
![E _ {{ij}} \ cdot E _ {{kl}} = {\ begin {cases} E _ {{il}} & {\ text {if}} j = k \\ 0 & {\ text {otherwise,}} \ end {cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aab8b73a088240e6ba287539b27230e80567b00)
where the zero matrix is the size .
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![m \ times p](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e459f087ef822dc6fba54b953c60de61be69c42)
Parameters
The following applies to the rank of a standard matrix
-
.
The same applies to the determinant and the trace of a square standard matrix
![m \ times m](https://wikimedia.org/api/rest_v1/media/math/render/svg/367523981d714dcd9214703d654bfdedbe58d44a)
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and .![\ operatorname {spur} (E _ {{ij}}) = \ delta _ {{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ef7dff803f54172c9b74d7b6d761f9f7348962)
The characteristic polynomial of a square standard matrix over a body is given by
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ chi (\ lambda) = {\ begin {cases} \ lambda ^ {{n-1}} (\ lambda -1) & {\ text {if}} i = j \\\ lambda ^ {n} & { \ text {otherwise.}} \ end {cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd88d1ec4e5ecaf5de085647e013deb818a0236)
In the case is therefore the only eigenvalue . For there is also the eigenvalue with simple multiplicity and the associated eigenvector .
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![i = j](https://wikimedia.org/api/rest_v1/media/math/render/svg/706e0928b2bf0f24076b0c90bb20616ff2068343)
![egg}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3)
use
Matrix entries
With the help of standard matrices, individual matrix entries can also be displayed as a trace. Is , then applies
![E _ {{ji}} \ in R ^ {{n \ times m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c23ddc8e5ac83a78493e6288c39d586a33aa7897)
![A \ in R ^ {m \ times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b86f150f527b6ae03f33869a6a05fa139c943d)
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.
For the product of two matrices and applies accordingly
![A \ in R ^ {{m \ times p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98ab7eacbf468042ce5a31694095797889ee3d84)
![B \ in R ^ {{p \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddcf022983e3e4dea11a3d304659dec426047ae)
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.
Standard base
The set of standard matrices over a given field forms the standard basis for the vector space of the matrices. Each matrix can thus be defined as a linear combination of standard
matrices![\ {E _ {{ij}} \ in K ^ {{m \ times n}} \ mid i = 1, \ ldots, m, j = 1, \ ldots, n \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7cc6895846b16fa8996f15276d5522e9141eec)
![A \ in K ^ {m \ times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab303433d6aa889f1f94258f6afb6a04ffea388d)
![A = \ sum _ {{i = 1}} ^ {m} \ sum _ {{j = 1}} ^ {n} a _ {{ij}} E _ {{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ee0262e472daf6385d5d70384f1b2aee91b6f6)
with represent. Thus, the four standard dies form , , and the standard basis of the area of matrices and are obtained for example
![a _ {{ij}} \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f96a8523bede0f971a412a30e9329258b366e12)
![E _ {{11}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcf510f6a311251752a33ff415a05eada90ca7d)
![E _ {{12}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3cd3f6da6648dfca7acb94b6a48617c93fd6b2)
![E _ {{21}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/780267746007165bb37cf5785f6fa309cdecb539)
![E _ {{22}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9635619bbd7a831775c5e00bf38151994264d6ba)
![(2 \ times 2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd967d734835dc2bf4d3f1b10707f0052a78a650)
-
.
Elementary matrices
Standard matrices are also used to represent the three types of elementary matrices of the shape
![{\ begin {aligned} R _ {{ij}} (\ alpha) & = I + \ alpha E _ {{ij}} \\ S_ {i} (\ gamma) & = I + (\ gamma -1) E _ {{ii }} \\ T _ {{i, j}} & = I-E _ {{ii}} - E _ {{jj}} + E _ {{ij}} + E _ {{ji}} \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/019801f6f5f0ba87ccda901f0dd25549b2ac0cb9)
with as the identity matrix and used. By multiplying from the left with such an elementary matrix, row operations, scalings and transpositions are carried out on a given matrix. These elementary matrices are used to describe the Gaussian elimination method for solving systems of linear equations .
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![\ alpha, \ gamma \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/a044e591261f9554bc218f27c52a5587511226af)
literature
- Tilo Arens, Frank Hettlich, Christian Karpfinger, Ulrich Kockelkorn, Klaus Lichtenegger, Hellmuth Stachel : Mathematics . 2nd Edition. Spektrum Akademischer Verlag, 2011, ISBN 3-8274-2347-3 .
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Michael Artin : Algebra . Springer, 1998, ISBN 3-7643-5938-2 .
- Christian Voigt, Jürgen Adamy: Collection of formulas for matrix calculation . Oldenbourg, 2007, ISBN 3-486-58350-6 .
Individual evidence
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^ Voigt, Adamy: Collection of formulas for matrix calculation . S. 8 .
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^ Arens et al: Mathematics . S. 508 .
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^ Artin: Algebra . S. 11 .