Matrix Transpose Calculator | Transpose Matrix Online
Calculate the transpose of a matrix easily online.
How to Use
- Enter the elements of the matrix in the text area.
- Use spaces or commas to separate numbers in a row, and use new lines for each row.
- Example: 1 2 3 4 5 6
- Click 'Calculate Transpose' to view the swapped rows and columns.
About Matrix Transposition
The transpose of a matrix is an operator which flips a matrix over its diagonal, switching the row and column indices of the matrix.
Definition
The transpose of an m×n matrix A, denoted as Aᵀ, is an n×m matrix. The element at the i-th row and j-th column of Aᵀ is equal to the element at the j-th row and i-th column of A.
Symmetric Matrices
A square matrix that is equal to its transpose (A = Aᵀ) is called a symmetric matrix. For symmetric matrices, the elements are totally symmetric across the main diagonal.
Properties of Transpose
- (Aᵀ)ᵀ = A
- (A + B)ᵀ = Aᵀ + Bᵀ
- (cA)ᵀ = cAᵀ, where c is a scalar
- (AB)ᵀ = BᵀAᵀ (Notice the reversed order of multiplication)
Applications
- Solving systems of linear equations.
- Data analysis techniques like Principal Component Analysis (PCA).
- Representing graphs and networks in computer science as adjacency matrices.