What does transposing a matrix mean?

Transposing a matrix means swapping rows and columns. For an m×n matrix A, the transpose Aᵀ is an n×m matrix where the entry in row i and column j of A becomes the entry in row j and column i of Aᵀ.

What are the dimensions of the transpose of an m×n matrix?

If A is an m×n matrix, then its transpose Aᵀ is an n×m matrix. The number of rows and columns are swapped.

How is the transpose used in linear algebra and data science?

The transpose appears in many contexts: defining symmetric and orthogonal matrices, expressing inner products, writing least squares solutions, reshaping datasets where rows represent observations and columns represent features, and implementing algorithms in linear algebra libraries.

Is transpose the same as conjugate transpose?

No. The transpose only swaps rows and columns. The conjugate transpose (or Hermitian transpose) also takes the complex conjugate of each entry. For real-valued matrices, the transpose and conjugate transpose coincide.

Matrix Transpose Calculator (Aᵀ)

Build an m×n matrix, compute its transpose Aᵀ, and get a clean numeric table plus a ready-to-copy LaTeX representation. Ideal for linear algebra homework, coding, and documentation.

Supports custom size, rectangular matrices, random fill, identity/zeros presets, and copying the result back into the input grid.

Enter a matrix and compute its transpose

Rows (m)

Columns (n)

Maximum size: 10×10. Use the buttons below to quickly fill common patterns (identity, zeros, random).

Input matrix A (m×n)

What is the transpose of a matrix?

Given an m×n matrix \(A\), its transpose, denoted by \(A^T\) (or sometimes \(A'\)), is the n×m matrix obtained by swapping rows and columns. Formally:

\[ (A^T)_{ij} = A_{ji} \]

In other words, the entry in the i-th row and j-th column of the transpose comes from the j-th row and i-th column of the original matrix.

Example

Start from the 2×3 matrix:

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]

The transpose is a 3×2 matrix:

\[ A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]

Every row of \(A\) becomes a column of \(A^T\), and every column of \(A\) becomes a row of \(A^T\).

Key properties of the transpose

\((A^T)^T = A\)
\((A + B)^T = A^T + B^T\)
\((cA)^T = cA^T\) for any scalar \(c\)
\((AB)^T = B^T A^T\) whenever the product \(AB\) is defined

These identities are fundamental in linear algebra, especially when working with inner products, adjoint operators, and least squares problems.

Why is the transpose useful?

Symmetric matrices: matrices for which \(A^T = A\), central in optimization and statistics.
Orthogonal matrices: where \(A^{-1} = A^T\), modeling rotations and rigid transformations.
Data manipulation: switching between “samples as rows” and “features as rows”.
Programming: many APIs and algorithms expect a particular orientation of vectors and matrices; transpose adapts data accordingly.

FAQ – Matrix transpose

What happens to the determinant when I transpose a matrix?

For a square matrix, the determinant of the transpose equals the determinant of the original matrix: \(\det(A^T) = \det(A)\). This is consistent with the idea that the transpose does not change the eigenvalues, only the orientation of the basis.

Does transposing change the eigenvalues of a matrix?

No. A matrix and its transpose have the same characteristic polynomial, so they share the same eigenvalues (with the same algebraic multiplicities).

How is transpose implemented in numerical libraries?

In low-level libraries, the transpose is often implemented either by changing the way indices are interpreted (strides) or by explicitly copying data into a new array with swapped axes. High-level libraries like NumPy, MATLAB, and Wolfram Language expose convenient operators such as A' or A.T that internally rely on these mechanisms.