Matrix Transpose
Matrix (one row per line)
The transpose of a matrix swaps its rows and columns. If A is an m×n matrix, Aᵀ is an n×m matrix. Transposes are used in linear algebra, machine learning, and solving systems of equations.
- 1(Aᵀ)ᵢⱼ = Aⱼᵢ
- 2Rows become columns
- 3(Aᵀ)ᵀ = A
- 4(AB)ᵀ = BᵀAᵀ
[[1,2,3],[4,5,6]]=[[1,4],[2,5],[3,6]]
Square matrix [[a,b],[c,d]]=[[a,c],[b,d]]
| Property | Formula |
|---|---|
| Double transpose | (Aᵀ)ᵀ = A |
| Sum | (A+B)ᵀ = Aᵀ+Bᵀ |
| Product | (AB)ᵀ = BᵀAᵀ |
| Scalar | (cA)ᵀ = cAᵀ |
| Symmetric matrix | A = Aᵀ |
References
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