Matrix Transpose
Flip rows into columns step by step · 전치행렬
1. What Is the Transpose of a Matrix?
The transpose of a matrix, written Aᵀ, is the matrix you get by flipping it over its main diagonal — every row becomes a column and every column becomes a row. The entry that was in row i, column j moves to row j, column i.
A practical consequence: an m×n matrix transposes into an n×m matrix. A wide 2×3 matrix becomes a tall 3×2 one. The transpose is one of the most common operations in linear algebra — it appears in dot products, least-squares fitting, and the definition of symmetric matrices.
전치행렬 Aᵀ는 주대각선을 기준으로 행렬을 뒤집은 것으로, 모든 행이 열이 되고 모든 열이 행이 됩니다. (i, j) 위치 원소는 (j, i)로 이동합니다. 따라서 m×n 행렬은 n×m 행렬이 됩니다.
2. The Transpose Rule
Each entry swaps its row and column index:
(Aᵀ)ᵢⱼ = aⱼᵢ
Worked example. For A = [[1, 2, 3], [4, 5, 6]] (a 2×3 matrix), the first row [1, 2, 3] becomes the first column, and the second row [4, 5, 6] becomes the second column:
Aᵀ = [[1, 4], [2, 5], [3, 6]] (a 3×2 matrix)
각 원소는 행과 열 인덱스를 맞바꿉니다: (Aᵀ)ᵢⱼ = aⱼᵢ. 예를 들어 2×3 행렬 [[1, 2, 3], [4, 5, 6]]의 전치는 3×2 행렬 [[1, 4], [2, 5], [3, 6]] 입니다.
3. The Reverse-Order Product Rule
The transpose of a sum is simple — it distributes: (A + B)ᵀ = Aᵀ + Bᵀ. But the transpose of a product reverses the order:
(AB)ᵀ = Bᵀ Aᵀ
The reversal is essential, not optional: BᵀAᵀ is the only arrangement whose dimensions line up. Applying the transpose twice returns the original matrix: (Aᵀ)ᵀ = A.
합의 전치는 (A + B)ᵀ = Aᵀ + Bᵀ로 분배되지만, 곱의 전치는 순서가 뒤집힙니다: (AB)ᵀ = BᵀAᵀ. 차원이 맞는 유일한 배열이기 때문입니다. 전치를 두 번 하면 원래 행렬로 돌아옵니다: (Aᵀ)ᵀ = A.
4. Symmetric Matrices & Key Properties
- (Aᵀ)ᵀ = A. Transposing twice undoes itself.
- (A + B)ᵀ = Aᵀ + Bᵀ. Distributes over addition.
- (AB)ᵀ = BᵀAᵀ. Reverses over multiplication.
- (cA)ᵀ = cAᵀ. A scalar passes straight through.
- Symmetric matrix. When A = Aᵀ, the matrix is symmetric — it mirrors across the diagonal. Only square matrices can be symmetric.
- det(Aᵀ) = det(A). The transpose has the same determinant as the original.
(Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ, (cA)ᵀ = cAᵀ 가 성립합니다. A = Aᵀ이면 대칭행렬(정사각 행렬만 가능)이고, det(Aᵀ) = det(A) 입니다.
5. Frequently Asked Questions
What does transposing a matrix do? It flips the matrix across its main diagonal, turning each row into a column. An m×n matrix becomes n×m.
What is (AB)ᵀ? It equals BᵀAᵀ — the transpose of a product reverses the order of the factors.
What is a symmetric matrix? A square matrix equal to its own transpose, A = Aᵀ, so the entries mirror across the main diagonal.
전치는 주대각선 기준으로 행렬을 뒤집어 행을 열로 바꿉니다(m×n → n×m). (AB)ᵀ = BᵀAᵀ로 순서가 뒤집힙니다. 대칭행렬은 A = Aᵀ인 정사각 행렬입니다.
Ready to practice? Drill the transpose and the rest of linear algebra on C:Matrix, or review the full matrix reference and the related matrix multiplication.