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Matrix Transpose

The transpose Aᵀ swaps rows and columns: (Aᵀ)[j,i] = A[i,j]. An m×n matrix becomes n×m. Symmetric matrices satisfy A = Aᵀ; skew-symmetric: A = −Aᵀ.

Concept Fundamentals
(Aᵀ)[j,i] = A[i,j]
Rule
m×n → n×m
Dimensions
A = Aᵀ
Symmetric
BᵀAᵀ
(AB)ᵀ

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Symmetric: A = Aᵀ; eigenvalues real. (AB)ᵀ = BᵀAᵀ (reverse order). AᵀA and AAᵀ are symmetric.

Key quantities
(Aᵀ)[j,i] = A[i,j]
Rule
Key relation
m×n → n×m
Dimensions
Key relation
A = Aᵀ
Symmetric
Key relation
BᵀAᵀ
(AB)ᵀ
Key relation

Ready to run the numbers?

Why: Transpose appears in least squares, Gram matrices, and symmetric eigenvalue problems. (AB)ᵀ = BᵀAᵀ.

How: For each (i,j), set (Aᵀ)[j,i] = A[i,j]. Column j of A becomes row j of Aᵀ.

Symmetric: A = Aᵀ; eigenvalues real.(AB)ᵀ = BᵀAᵀ (reverse order).
Sources:Gilbert Strang, Linear AlgebraKhan Academy

Run the calculator when you are ready.

Transpose MatrixSwap rows and columns; dimensions flip

Quick Examples — Click to Load

Matrix A(3×3)

matrix_transpose.sh
CALCULATED
$ transpose --from="3×3" --to="3×3"
Original
3×3
Transposed
3×3
Original A (3×3)
000
000
000
Transposed Aᵀ (3×3)
000
000
000
Share:
Matrix Transpose
3×3 → 3×3
[0, 0, 0] | [0, 0, 0] | [0, 0, 0]
numbervibe.com/calculators/mathematics/linear-algebra/matrix-transpose-calculator

Aᵀ Element Distribution (Bar)

Sign Distribution (Doughnut)

Calculation Steps

SETUP
Original A[0, 0, 0] [0, 0, 0] [0, 0, 0]
SETUP
Dimensions3x3
FORMULA
Transpose rule(A^T)[j,i] = A[i,j]
RESULT
A^T dimensions3x3
RESULT
Transposed A^T[0, 0, 0] [0, 0, 0] [0, 0, 0]

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

↔

(Aᵀ)ᵀ = A

📐

det(Aᵀ) = det(A)

Key Takeaways

  • • Transpose swaps rows and columns: (Aᵀ)[j,i] = A[i,j].
  • • m×n matrix → n×m result. Dimensions flip.
  • • (Aᵀ)ᵀ = A — double transpose returns original.
  • • (AB)ᵀ = BᵀAᵀ — order reverses for product.
  • • Symmetric: A = Aᵀ. Skew-symmetric: A = -Aᵀ.

Did You Know?

⚡Transpose is O(mn)—one pass, swap indices.Source: Complexity
🧠Neural nets use (Wx)ᵀ for backprop—transpose of weight matrix.Source: ML
📊Covariance matrix Σ = XᵀX/n uses transpose.Source: Stats
🎮Graphics: transpose for column-major vs row-major layouts.Source: Graphics
⏱️det(Aᵀ) = det(A). Eigenvalues of Aᵀ = eigenvalues of A.Source: Properties
⚛️Quantum: conjugate transpose A* for Hermitian adjoint.Source: Quantum

How It Works

Swap row and column indices: (Aᵀ)[j,i] = A[i,j]. Row i of A becomes column i of Aᵀ.

A (m×n) → Aᵀ (n×m)

Element at (i,j) moves to (j,i).

Expert Tips

Dimensions Flip

m×n → n×m. Row vector becomes column vector.

(AB)ᵀ = BᵀAᵀ

Product transpose: reverse order of matrices.

Symmetric Check

A = Aᵀ means a[i,j] = a[j,i]. Diagonal is free.

Skew-Symmetric

A = -Aᵀ: diagonal zeros, a[i,j] = -a[j,i].

Comparison Table

FeatureThis CalculatorNumPyManual
Original vs transposed✅❌⚠️
Bar & Doughnut charts✅❌❌
8 preset examples✅❌❌
Step-by-step✅❌⚠️

FAQ

What does transpose do?

Swaps rows and columns. (Aᵀ)[j,i] = A[i,j].

Do dimensions change?

Yes. m×n → n×m.

Is (Aᵀ)ᵀ = A?

Yes. Transposing twice returns the original.

What is (AB)ᵀ?

BᵀAᵀ. The order of matrices reverses.

What is a symmetric matrix?

A = Aᵀ. Square matrix with a[i,j] = a[j,i].

What is skew-symmetric?

A = -Aᵀ. Diagonal zeros, a[i,j] = -a[j,i].

Does det(Aᵀ) = det(A)?

Yes. Transpose preserves determinant.

Eigenvalues of A vs Aᵀ?

Same eigenvalues. Eigenvectors may differ.

Stats

m×n→n×m
Dimensions
O(mn)
Complexity
(Aᵀ)ᵀ=A
Involution
(AB)ᵀ=BᵀAᵀ
Product

Sources

Disclaimer: For educational purposes. Uses JavaScript floating-point. Verify critical calculations independently.

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