Kaip apskaičiuoti Matrix Transpose
Kas yra Matrix Transpose?
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.
Formulė
If A is m × n, then Aᵀ is n × m where Aᵀ[i,j] = A[j,i]
- A
- original matrix
- Aᵀ
- transposed matrix
Žingsnis po žingsnio vadovas
- 1(Aᵀ)ᵢⱼ = Aⱼᵢ
- 2Rows become columns
- 3(Aᵀ)ᵀ = A
- 4(AB)ᵀ = BᵀAᵀ
Worked Examples
Įvestis
[[1,2,3],[4,5,6]]
Rezultatas
[[1,4],[2,5],[3,6]]
Įvestis
Square matrix [[a,b],[c,d]]
Rezultatas
[[a,c],[b,d]]
Frequently Asked Questions
What is the transpose of a transpose?
(Aᵀ)ᵀ = A. Transposing twice returns the original.
Is the transpose of a product the product of transposes in reverse?
Yes! (AB)ᵀ = BᵀAᵀ (note the reversal).
What is a symmetric matrix?
A square matrix where A = Aᵀ. The matrix equals its transpose.
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