Matrix Transpose Calculator
Calculate the transpose of a matrix (flip matrix over its diagonal).
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Matrix Transpose
The transpose of a matrix A is obtained by flipping the matrix over its diagonal:
\[ (A^T)_{ij} = A_{ji} \]
Properties:
- \((A^T)^T = A\)
- \((A + B)^T = A^T + B^T\)
- \((cA)^T = cA^T\) for scalar c
- \((AB)^T = B^TA^T\)
- \(\det(A^T) = \det(A)\)
Special Types of Matrices:
- Symmetric: \(A^T = A\)
- Skew-symmetric: \(A^T = -A\)
- Orthogonal: \(A^TA = AA^T = I\)
- Hermitian: \(A^T = \overline{A}\)
Applications:
- Solving systems of linear equations
- Least squares regression
- Computer graphics transformations
- Quantum mechanics operators
- Machine learning algorithms
Geometric Interpretation:
- Reflection over the main diagonal
- Preserves inner products
- Changes row space to column space
- Maintains matrix rank