Matrix Transpose Calculator

Enter any m×n matrix to swap its rows and columns and get the transpose Aᵀ, along with its new dimensions, Frobenius norm, and a symmetry check.

Quick Facts

Transpose definition
(Aᵀ)ᵢⱼ = Aⱼᵢ
Row i, column j of A becomes row j, column i of Aᵀ.
Shape flip
m × n → n × m
An m-row, n-column matrix becomes n rows and m columns.
Double transpose
(Aᵀ)ᵀ = A
Transposing twice always returns the original matrix.
Symmetric matrices
A = Aᵀ
Only square matrices can equal their own transpose.

Your Results

Calculated
Transposed Matrix Aᵀ
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(Aᵀ)ᵢⱼ = Aⱼᵢ, rows and columns swapped
Dimensions
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Original size → transposed size
Frobenius Norm
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‖A‖_F = √(Σ aᵢⱼ²), unchanged by transposing
Symmetric?
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Whether A equals its own transpose

Ready

Enter matrix rows (comma or space separated), then press Calculate.

How the Matrix Transpose Calculator works

The transpose of a matrix A, written AT, is formed by turning A's rows into columns (and its columns into rows). Formally, if A is an m×n matrix, its transpose AT is the n×m matrix defined entry by entry as (AT)ij = Aji, where i is the row index and j is the column index. This calculator accepts any rectangular matrix, swaps its rows and columns, and reports the transposed matrix along with its new dimensions, Frobenius norm, and whether the matrix is symmetric.

Formula and worked example

To transpose a matrix, the first row of A becomes the first column of AT, the second row becomes the second column, and so on. For example, with the 3×2 matrix A = [[1, 2], [3, 4], [5, 6]]:

AT = [[1, 3, 5], [2, 4, 6]], a 2×3 matrix. Entry A21 = 3 (row 2, column 1 of A) moved to position (1, 2) in AT, matching the rule (AT)12 = A21. Notice that no arithmetic is performed on the values — transposing only rearranges their positions, which is why quantities like the sum of squares of all entries (and therefore the Frobenius norm) stay exactly the same before and after transposing.

Dimensions, symmetry, and the double transpose

Transposing always flips the shape: an m×n matrix becomes n×m. Only a square matrix (m = n) can keep the same dimensions after transposing, and even then the entries usually move to new positions unless the matrix is symmetric, meaning A = AT (equivalently, Aij = Aji for every i and j). Transposing twice always restores the original matrix: (AT)T = A. These properties make the transpose useful for quickly checking a matrix's structure — a non-square matrix can never be symmetric, and comparing A to AT entry by entry is the standard way to test for symmetry.

Common sources of error

  • Mismatched row lengths: every row you enter must have the same number of values — a ragged matrix (e.g., one row with 2 numbers and another with 3) is not valid and cannot be transposed.
  • Confusing transpose with inverse: the transpose only rearranges entries and never changes a matrix's values; the inverse A-1 (which only exists for certain square matrices) is a completely different operation, even though for orthogonal matrices AT happens to equal A-1.
  • Misreading indices: remember (AT)ij = Aji, not Aij — the row and column indices swap, so it is easy to transpose an entry into the wrong position by hand on larger matrices.

Real-world applications

The transpose is one of the most-used operations in linear algebra: it converts a column vector into a row vector (and back) so vectors and matrices can be multiplied correctly; it appears in the normal equations ATAx = ATb used to solve least-squares regression problems; it is used to build the covariance matrix in statistics (Σ = (1/n) XTX for centered data); and testing whether AT = A-1 identifies orthogonal matrices, which represent rotations and reflections that preserve length in geometry and computer graphics.

Frequently Asked Questions

What does it mean to transpose a matrix?
Transposing a matrix means swapping its rows and columns: the entry in row i, column j of the original matrix A becomes the entry in row j, column i of the transpose AT. In symbols, (AT)ij = Aji. Visually, the first row of A becomes the first column of AT, the second row becomes the second column, and so on.
Does the size of a matrix change when you transpose it?
Yes, unless the matrix is square. An m×n matrix (m rows, n columns) becomes an n×m matrix after transposing. For example, a 3×2 matrix becomes a 2×3 matrix. For a square n×n matrix, the transpose is also n×n, though the entries generally still move to different positions.
What is a symmetric matrix?
A symmetric matrix is a square matrix that equals its own transpose: A = AT, which means Aij = Aji for every i and j. Only square matrices can be symmetric, since transposing a non-square matrix always changes its dimensions.
What are common uses of the matrix transpose?
The transpose appears throughout linear algebra: converting a row vector to a column vector (and back), forming the normal equations ATAx = ATb used in least-squares regression, checking whether a matrix is symmetric or orthogonal (where AT = A-1), and computing dot products and covariance matrices in statistics and machine learning.

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