Matrix Transpose Calculator
Transpose matrices of any size by swapping rows and columns.
Matrix A (3×3):
What Is the Matrix Transpose Calculator?
The Matrix Transpose Calculator swaps the rows and columns of any matrix instantly. Enter your matrix using the visual grid, choose dimensions from 2×2 up to 5×5, and the transposed result appears side-by-side with your input. For square matrices, the calculator also checks whether the matrix is symmetric (A = Aᵀ) or skew-symmetric (A = −Aᵀ), and displays the trace.
- ›Flexible dimensions: independently set rows and columns from 2 to 5, supporting rectangular matrices from 2×3 to 5×4 and everything in between.
- ›Visual bracket notation: the input matrix A and result Aᵀ are displayed with bracket notation side-by-side so the row-to-column swap is immediately visible.
- ›Symmetry detection: for square matrices, instantly shows whether A = Aᵀ (symmetric) or A = −Aᵀ (skew-symmetric).
- ›Trace display: for square matrices, the common trace of A and Aᵀ is computed (tr(A) = tr(Aᵀ) always).
- ›Copy to clipboard: copy the transposed matrix as tab-separated rows for pasting into spreadsheets or code.
Formula
| Property | Formula | Notes |
|---|---|---|
| Definition | Aᵀ[i][j] = A[j][i] | Swap every row index with column index |
| Double transpose | (Aᵀ)ᵀ = A | Transposing twice returns the original |
| Sum rule | (A + B)ᵀ = Aᵀ + Bᵀ | Transpose distributes over addition |
| Product rule | (AB)ᵀ = BᵀAᵀ | Order reverses for matrix products |
| Scalar rule | (kA)ᵀ = k Aᵀ | Scalars pass through unchanged |
| Determinant | det(Aᵀ) = det(A) | Square matrices only |
How to Use
- 1Select the number of rows using the Rows buttons (2–5).
- 2Select the number of columns using the Columns buttons (2–5).
- 3Enter each matrix entry in the grid (blank cells default to 0).
- 4Click "Transpose", or press Enter from any cell, to compute Aᵀ.
- 5Read the result matrix Aᵀ shown to the right of the arrow.
- 6For square matrices, check the symmetry and trace cards below the matrices.
- 7Click "Copy Aᵀ" to copy the transposed matrix as tab-separated text.
Example Calculation
Example 1, Transpose a 2×3 matrix
Example 2, Symmetric matrix check
Why the product rule reverses order
Understanding Matrix Transpose
Symmetric and Skew-Symmetric Matrices
A square matrix A is symmetric if A = Aᵀ, meaning every entry satisfies A[i][j] = A[j][i]. Symmetric matrices appear throughout physics (inertia tensors, stress tensors), statistics (covariance matrices), and graph theory (adjacency matrices of undirected graphs). They have exclusively real eigenvalues and can always be orthogonally diagonalized.
A matrix is skew-symmetric (anti-symmetric) if A = −Aᵀ, meaning A[i][j] = −A[j][i] for all i, j. This forces the diagonal entries to be zero (since A[i][i] = −A[i][i] implies A[i][i] = 0). Skew-symmetric matrices arise in physics as representations of angular velocity and electromagnetic field tensors.
- ›Every square matrix can be decomposed uniquely as A = S + K where S = (A + Aᵀ)/2 is symmetric and K = (A − Aᵀ)/2 is skew-symmetric.
- ›The sum of two symmetric matrices is symmetric; the product is symmetric only if the matrices commute.
- ›All eigenvalues of a real symmetric matrix are real (Spectral Theorem).
- ›All eigenvalues of a real skew-symmetric matrix are purely imaginary or zero.
The Trace is Invariant Under Transposition
The trace of a square matrix, the sum of its diagonal entries, is preserved by transposition: tr(Aᵀ) = tr(A). This follows directly from the definition: the diagonal entries A[i][i] are unchanged when rows and columns are swapped. The trace is also invariant under cyclic permutations of products: tr(ABC) = tr(BCA) = tr(CAB).
- ›tr(A + B) = tr(A) + tr(B), trace is linear.
- ›tr(AB) = tr(BA) even when AB ≠ BA, trace commutes.
- ›tr(Aᵀ) = tr(A), transposition preserves trace.
- ›The Frobenius inner product of two matrices is tr(AᵀB) = Σᵢⱼ Aᵢⱼ Bᵢⱼ.
Transpose in Linear Algebra and Applications
The transpose operation is fundamental to numerous areas of linear algebra and its applications:
- ›Least squares: the normal equations for Ax = b are AᵀAx = Aᵀb, the transpose of the coefficient matrix appears naturally in the least-squares solution.
- ›Orthogonal matrices: a matrix Q is orthogonal iff QᵀQ = I, meaning Q⁻¹ = Qᵀ. Rotations and reflections are orthogonal matrices.
- ›SVD decomposition: the singular value decomposition A = UΣVᵀ expresses any matrix as a product involving the transpose of V.
- ›Gram matrix: AᵀA (for real matrices) and A*A (for complex matrices) are always positive semi-definite, used in kernel methods and PCA.
- ›Neural networks: the transpose of the weight matrix appears in backpropagation to compute gradients with respect to layer inputs.
Rectangular Matrices and Dimension Flip
For a non-square m×n matrix A, the transpose Aᵀ is n×m, dimensions are flipped. This means the transpose of a column vector (n×1 matrix) is a row vector (1×n), and vice versa. In the context of inner products, the dot product of two vectors u and v can be written as uᵀv (treating them as column vectors), a notation that extends naturally to quadratic forms xᵀAx.
- ›A column vector v ∈ ℝⁿ is an n×1 matrix; its transpose vᵀ is a 1×n row vector.
- ›The outer product vvᵀ is an n×n rank-1 matrix; the inner product vᵀv is a scalar.
- ›
- ›
Frequently Asked Questions
What is the transpose of a matrix?
The transpose of a matrix A, written Aᵀ, is formed by swapping its rows and columns: the entry in row i and column j of Aᵀ equals the entry in row j and column i of A. If A has dimensions m×n, then Aᵀ has dimensions n×m.
Intuitively, the first row of A becomes the first column of Aᵀ, the second row becomes the second column, and so on. For a 2×3 matrix:
What does it mean for a matrix to be symmetric?
A square matrix A is symmetric if A = Aᵀ, equivalently, every entry satisfies A[i][j] = A[j][i]. Visually, the matrix is a mirror image of itself across the main diagonal.
- ›Covariance matrices in statistics are always symmetric positive semi-definite.
- ›The adjacency matrix of an undirected graph is symmetric.
- ›All eigenvalues of a real symmetric matrix are real (Spectral Theorem).
- ›Real symmetric matrices can always be diagonalized by an orthogonal matrix.
A related concept is skew-symmetry: A = −Aᵀ. Skew-symmetric matrices have zeros on the diagonal and A[i][j] = −A[j][i] for i ≠ j.
Why does the product rule for transposes reverse order: (AB)ᵀ = BᵀAᵀ?
This is because matrix multiplication is not commutative and the index structure forces the reversal. The (i, j) entry of AB is the dot product of row i of A with column j of B. When you transpose, this entry moves to position (j, i).
To express this as a product, you need: row j of Bᵀ (which was column j of B) dotted with column i of Aᵀ (which was row i of A). That is exactly the (j, i) entry of BᵀAᵀ, not AᵀBᵀ.
A useful way to remember this: think of reversing the order of putting on socks and shoes. To undo (AB), you first undo B (apply B⁻¹), then undo A (apply A⁻¹). The same reversal applies to the transpose.
Is the transpose of a product of three matrices (ABC)ᵀ = CᵀBᵀAᵀ?
Yes, the rule extends to any number of factors. The transpose of a product reverses the order of all factors: (ABC)ᵀ = CᵀBᵀAᵀ.
This follows by applying the two-matrix rule twice:
- ›(ABC)ᵀ = ((AB)C)ᵀ = Cᵀ(AB)ᵀ = Cᵀ(BᵀAᵀ) = CᵀBᵀAᵀ
- ›For inverses of products: (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹, the same order-reversal applies.
- ›For four matrices: (ABCD)ᵀ = DᵀCᵀBᵀAᵀ.
How is the transpose used in computing the matrix inverse?
For orthogonal matrices (Q such that QQᵀ = I), the inverse equals the transpose: Q⁻¹ = Qᵀ. This makes computing the inverse trivially fast, just swap rows and columns, and is why rotation matrices are so convenient to work with.
For general invertible matrices, the transpose appears in the adjugate formula: A⁻¹ = (1/det(A)) · adj(A), where adj(A) = Cᵀ and C is the matrix of cofactors. Computing the transpose of the cofactor matrix is the final step in this inverse formula.
- ›Orthogonal matrix: Q⁻¹ = Qᵀ (very efficient, just a transpose).
- ›General inverse: A⁻¹ = adj(A)/det(A) where adj(A) = cofactor matrix transposed.
- ›For 2×2: [[a,b],[c,d]]⁻¹ = (1/(ad-bc)) · [[d,−b],[−c,a]], note the element swap is related to transposition of cofactors.
What happens when you multiply a matrix by its transpose?
The product AᵀA (or AAᵀ) always produces a symmetric, positive semi-definite matrix. This has important consequences:
- ›AᵀA is always square (n×n if A is m×n) and symmetric: (AᵀA)ᵀ = AᵀA.
- ›All eigenvalues of AᵀA are non-negative (positive if A has full column rank).
- ›The singular values of A are the square roots of the eigenvalues of AᵀA.
- ›The Gram matrix AᵀA encodes all inner products between columns of A.
- ›The normal equations for least squares are AᵀAx = Aᵀb.
AAᵀ (m×m) and AᵀA (n×n) share the same nonzero eigenvalues, which is the foundation of the singular value decomposition (SVD).
Does transposing a matrix change its determinant?
No, for any square matrix, det(Aᵀ) = det(A). The determinant is unchanged by transposition. This follows from the Leibniz formula: the determinant sums over all permutations of column indices, and transposing corresponds to swapping which index (row vs column) is permuted, giving the same sum.
Consequences of det(Aᵀ) = det(A):
- ›A matrix and its transpose are simultaneously singular or simultaneously invertible.
- ›The characteristic polynomials of A and Aᵀ are identical: det(A − λI) = det(Aᵀ − λI).
- ›The eigenvalues of A and Aᵀ are the same (though eigenvectors generally differ).
- ›For orthogonal Q: det(Q) = ±1 since det(Q)·det(Qᵀ) = det(QQᵀ) = det(I) = 1.