Antisymmetric matrix
An antisymmetric matrix is a square matrix A whose transpose is equal to its opposite, that is, the relation is valid AT = -A.
An array of m × n elements (m = rows, n = columns):
- A=[chuckles]a11a12a13 a1na21a22a23 a2na31a32a33 a3n am1am2am3 amn]###### ##########################################################################################################################################################################################################################################################
![{displaystyle A=left[{begin{array}{ccccc}a_{11}&a_{12}&a_{13}&cdots &a_{1n}\a_{21}&a_{22}&a_{23}&cdots &a_{2n}\a_{31}&a_{32}&a_{33}&cdots &a_{3n}\vdots &vdots &vdots &ddots &vdots \a_{m1}&a_{m2}&a_{m3}&cdots &a_{mn}\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f170a4f492d829eafd92e2515ac2e74f43c8fa)
is antisimetric (or hemisimetric), if it is a square matrix (m = n) and aji=− − aij{displaystyle a_{ji}=-a_{ij}}} for everything i, j =1,2,3,...n. Accordingly, aii=0{displaystyle a_{ii}=0} for everything i. Therefore, the matrix A assumes the form:
- A=[chuckles]0a12a13 a1n− − a120a23 a2n− − a13− − a230 a3n − − a1n− − a2n− − a3n 0]################################################################################################ ################################################################################################################################################################
![{displaystyle A=left[{begin{array}{ccccc}0&a_{12}&a_{13}&cdots &a_{1n}\-a_{12}&0&a_{23}&cdots &a_{2n}\-a_{13}&-a_{23}&0&cdots &a_{3n}\vdots &vdots &vdots &ddots &vdots \-a_{1n}&-a_{2n}&-a_{3n}&cdots &0\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d31f2c20ee391891d272182e48057d3bca2da539)
Example
The Matrix
- A=[chuckles]0− − 24202− − 4− − 20]{displaystyle A=left[{begin{array}{rr}{0}{0}{0}{4}{4{2}{2}{2}{2}{{{4}{{{}{{{4}{{}{{{{}{{{{}{{{}{{}{{{{}{}{}{{}{{{{}{{{{{{}{}{{{{}{}{}{{}{{}{}{}{{{{}{{{{}{}{}{{{{}{{{{{}{{}{}{}{}{{{}{}{{{}{{{}{}{}{
![{displaystyle A=left[{begin{array}{rrr}{0}&{-2}&{4}\{2}&{0}&{2}\{-4}&{-2}&{0}\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281db94e6b075ed44629e59a1a549d42e98be7eb)
is antisymmetric, since
- AT=[chuckles]02− − 4− − 20− − 2420]=− − A{displaystyle A^{T}=left[{begin{array}{rr}{rr}{0}{4}{2}{{{-2}{{0}{0}{{{{4}{{4}{{}{{{}{{{{{}{{{}{}{{{{}{{}{{{{}{{{{}{{{}{{{}{{{{}{}{{}{{{}{{{}{{{{}{}{}{{{{}{{{{{{}{}{{}{{{}{}{{{{}{{{}{}{{
![{displaystyle A^{T}=left[{begin{array}{rrr}{0}&{2}&{-4}\{-2}&{0}&{-2}\{4}&{2}&{0}\end{array}}right]=-A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a16afdf6907434e9811ae77c0399fbdd4fadf9ba)
The main diagonal is preserved and all other numbers are changed to the opposite sign. Note that the transpose matrix of the antisymmetric matrix A is -A, and that the antisymmetry is with respect to the main diagonal.
If n=m is odd, the determinant of the matrix will always be 0
Symmetric and antisymmetric matrix decomposition
Let A be a square matrix, it can be decomposed into a sum of symmetric and antisymmetric parts as follows:
- A=12(A+AT)+12(A− − AT){displaystyle A={frac {1}{2}}}left(A+A^{Tright)+{frac {1}{2}}}left(A-A^{T}right)}
where the antisymmetric part is
- 12(A− − AT){displaystyle {frac {1}{2}}left(A-A^{T}right)}
| Demonstration |
| Transposition properties are used.
(12(A− − AT))T=12(A− − AT)T=12(AT− − (AT)T)=12(AT− − A)=− − 12(A− − AT){cHFFFFFF}{cH00FFFF}{cHFFFFFF00}{cHFFFFFF00}{cHFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF00}{cHFFFFFF00}{cHFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{c}{cH00}{cHFFFFFFFFFFFFFFFFFF It is then proved by definition that 12(A− − AT){displaystyle {frac {1}{2}}left(A-A^{T}right)} It's antisymmetric. |
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