1. What is Matrix Transpose?

The transpose of a matrix is a new matrix whose rows are the columns of the original matrix. For any matrix A, the transpose A^T is defined by A^T_ij = A_ji.

2. How Does the Calculator Work?

The calculator uses the matrix transpose formula:

Where:

• \( A \) — Original matrix
• \( A^T \) — Transposed matrix
• \( i,j \) — Row and column indices

Explanation: The calculator swaps rows and columns of the input matrix to produce the transposed result.

3. Importance of Matrix Transpose

Details: Matrix transposition is fundamental in linear algebra, used in solving systems of equations, matrix decompositions, and various applications in physics, engineering, and computer science.

4. Using the Calculator

Tips: Enter your matrix with each row on a new line and elements separated by spaces or commas. The calculator will automatically compute the transpose.

5. Frequently Asked Questions (FAQ)

Q1: What are the properties of matrix transpose?
A: Key properties include: (A^T)^T = A, (A+B)^T = A^T+B^T, (AB)^T = B^TA^T

Q2: What's the difference between transpose and inverse?
A: Transpose swaps rows and columns, while inverse is a matrix that when multiplied with the original gives the identity matrix.

Q3: What is a symmetric matrix?
A: A matrix that equals its own transpose (A = A^T).

Q4: Can rectangular matrices be transposed?
A: Yes, transposition works for any m×n matrix, resulting in an n×m matrix.

Q5: How is transpose used in machine learning?
A: Frequently used in operations like computing gradients, solving normal equations, and various matrix factorizations.