Matrix Transpose Calculator — with Symmetry Check

Q: What is the transpose of a matrix?

The transpose of a matrix A, written A^T, is obtained by swapping rows and columns. If A is an m×n matrix, then A^T is n×m. Element A^T[i][j] = A[j][i].

Q: What is a symmetric matrix?

A square matrix A is symmetric if A = A^T, meaning A[i][j] = A[j][i] for all i and j. Covariance matrices in statistics are always symmetric. So are many matrices in physics (moment of inertia, stress tensors).

Q: What are the properties of transpose?

Key properties: (A^T)^T = A; (A + B)^T = A^T + B^T; (kA)^T = k·A^T; (AB)^T = B^T·A^T (note the reversed order!); det(A^T) = det(A); rank(A^T) = rank(A).

Q: How is transpose used in machine learning?

Transpose is used constantly in ML: computing dot products (a^T b), forming Gram matrices (X^T X), transposing weight matrices in backpropagation, computing covariance matrices (X^T X / n), and in attention mechanisms (Q K^T).

Q: What is the difference between transpose and inverse?

Transpose swaps rows and columns; inverse solves A·A⁻¹ = I. For orthogonal matrices (like rotation matrices), the transpose equals the inverse: A^T = A⁻¹. These are called orthogonal matrices and are computationally efficient because transposing is much cheaper than inverting.

Matrix Transpose Calculator

Transpose any matrix from 2×2 up to 5×5 with full element-by-element steps. Detects symmetric matrices and verifies key transpose properties.

Quick Answer

A^T[i][j] = A[j][i]. A 3×4 matrix transposes to a 4×3 matrix. A matrix is symmetric if A = A^T (true for covariance matrices, Gram matrices, and rotation-symmetric tensors).

Matrix Dimensions

Rows

Columns

A (3×3) → A^T (3×3)

Matrix A (3×3)

What Is the Transpose?

The transpose of a matrix A is formed by writing its rows as columns. If A has shape m×n, then A^T has shape n×m. This operation is fundamental to linear algebra and appears in every branch of science and engineering.

Key Properties

(A^T)^T = A — transposing twice returns the original
(A + B)^T = A^T + B^T — transpose distributes over addition
(AB)^T = B^T A^T — order reverses when transposing a product
det(A^T) = det(A) — determinant is unchanged
rank(A^T) = rank(A) — rank is unchanged

Symmetric Matrices

A square matrix where A = A^T is called symmetric. Symmetric matrices have real eigenvalues and orthogonal eigenvectors (Spectral Theorem). They appear everywhere:

Statistics: Covariance matrices are always symmetric positive semi-definite
Physics: Inertia tensors, stress tensors, and diffusion tensors are symmetric
Graph theory: Adjacency matrices of undirected graphs are symmetric
Optimization: Hessian matrices of smooth functions are symmetric

International Notation

The transpose is written as A^T in most countries. Some European and Russian texts use A' (prime) or Aᵀ. In physics, the Hermitian conjugate (conjugate transpose) A† is used for complex matrices, common in quantum mechanics.

Orthogonal Matrices

For orthogonal matrices (rotation, reflection matrices), A^T = A⁻¹. This is enormously useful: transposing is O(n²) while general inversion is O(n³). Computer graphics, robotics, and computer vision exploit this to efficiently invert transformation matrices.

Frequently Asked Questions

What is the transpose of a matrix?

The transpose of A, written A^T, swaps rows and columns: A^T[i][j] = A[j][i]. A 3×4 matrix transposes to a 4×3 matrix.

What is a symmetric matrix?

A square matrix where A = A^T. This means A[i][j] = A[j][i] for all positions. Covariance matrices and Gram matrices (X^T X) are always symmetric.

What are the properties of transpose?

(A^T)^T = A; (A+B)^T = A^T + B^T; (AB)^T = B^T A^T; det(A^T) = det(A); rank(A^T) = rank(A).

How is transpose used in machine learning?

Constantly: dot products (a^T b), Gram matrices (X^T X), backpropagation weight gradients, covariance estimation, and transformer attention (Q K^T / √d).

What is the difference between transpose and inverse?

Transpose swaps rows/columns; inverse satisfies A A⁻¹ = I. For orthogonal matrices, A^T = A⁻¹ — making transpose a free inverse.

Related Calculators

Matrix Multiplication Calculator — use (AB)^T = B^T A^T
Matrix Inverse Calculator — compare with orthogonal matrices
Matrix Determinant — det(A^T) = det(A)
Gram-Schmidt Process — produces orthogonal (transposable) bases
Eigenvalue Calculator — symmetric matrices have real eigenvalues