Matrix Transpose Calculator

Calculate the transpose of any matrix quickly and easily

Matrix Operations

The transpose of a matrix is formed by turning rows into columns and columns into rows, creating a new matrix with dimensions flipped.

Understanding Matrix Transposition

Matrix transposition is a fundamental operation in linear algebra where the rows and columns of a matrix are interchanged. For a matrix A, its transpose is denoted as A^T or A'.

Properties of Matrix Transpose

The transpose of an m × n matrix is an n × m matrix
(A^T)^T = A (transpose of transpose is the original matrix)
(A + B)^T = A^T + B^T (transpose of sum equals sum of transposes)
(AB)^T = B^TA^T (transpose of product equals product of transposes in reverse order)

Example

For the matrix:

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]

Its transpose is:

\[ A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]

Applications

Matrix transposition is essential in:

Data analysis and statistics
Image processing and computer graphics
Machine learning algorithms
Signal processing
Optimization problems

Special Cases

Symmetric Matrices

A matrix is symmetric if A = A^T. Example:

\[ \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \]

Skew-Symmetric Matrices

A matrix is skew-symmetric if A = -A^T. Example:

\[ \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \]