Matrix Transpose Calculator
5), fractions (1/3), decimals (-2.5), and complex (2+3i).
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Matrix Analysis & Mathematical Invariants
\operatorname{tr}(A) = \operatorname{tr}(A^T). The sum of main diagonal elements remains identical.
\det(A) = \det(A^T). Transposition preserves the determinant for all square matrices.
How Matrix Transposition Works
To transpose an matrix, swap the row and column indices for every element: (Aᵀ)ᵢⱼ = Aⱼᵢ.
The rows of Matrix A become the columns of Matrix Aᵀ.
Square Matrix Decomposition Theorem
Any square matrix A can be uniquely decomposed into the sum of a Symmetric Matrix (S) and a Skew-Symmetric Matrix (K):
A = S + K = ½(A + Aᵀ) + ½(A - Aᵀ).
Symmetric decomposition is only defined for square matrices ($m = n$). Current size is .
Gramian Matrix Products (Gram Matrices)
The products G₁ = A · Aᵀ (size ) and G₂ = Aᵀ · A (size ) are always symmetric and positive semi-definite for any real matrix $A$.
Matrix Transpose Calculator
Transpose matrices of any dimension from 1×1 up to 8×8 instantly. Calculate standard matrix transposes, conjugate transposes (Hermitian adjoint), exact fractional values, symmetric matrix checks, trace invariance, determinant invariance, and Gramian products with full step-by-step mathematical explanations.
For any matrix \(A\), its double transpose returns the original matrix: \((A^T)^T = A\). Furthermore, transposition preserves the determinant and trace: \(\det(A^T) = \det(A)\) and \(\operatorname{tr}(A^T) = \operatorname{tr}(A)\).
