Matrix Transpose Calculator
Enter matrix size and values, optionally choose conjugate transpose, then compute instantly. See formatted equation preview and clear results below.
Equation Preview
Helping Notes
Required inputs mirror common transpose tools: matrix size (m×n), matrix entries, and an option to take the complex conjugate. :contentReference[oaicite:1]{index=1}
Empty cells count as zero. Complex entries are supported; conjugate transpose applies element-wise conjugation before transposing.
Results
Transposed Matrix
Formatted Output
Info
What Is a Matrix Transpose Calculator?
A Matrix Transpose Calculator converts any matrix into its transpose by swapping rows and columns. Each entry moves from row , column to row , column . This operation is fundamental in linear algebra, appearing in normal equations, least squares, orthogonality checks, covariance matrices, and vector space duals. The calculator supports rectangular, square, and block matrices, and can show related objects such as the conjugate transpose for complex inputs. With render‑ready formulas and clear steps, it’s ideal for learning, homework, and quick verification before coding or submitting solutions.
About the Matrix Transpose Calculator
The transpose keeps numerical values but changes their positions. If , then . It preserves many key properties: , , , and . Determinant and trace are unchanged for square matrices—, —and . For complex matrices, the conjugate transpose (Hermitian) is . Symmetric matrices satisfy ; skew‑symmetric satisfy . The calculator handles vectors (row ↔ column), ensures dimension compatibility, and optionally displays block‑transpose rules, which are vital when working with partitioned matrices.
Entrywise definition:
Shape:
Involutive:
Linear:
Product:
Square‑matrix properties:
Hermitian (complex):
How to Use This Matrix Transpose Calculator
- Paste or type your matrix. Use semicolons for rows or a grid input. Non‑numeric symbols are supported in exact mode.
- Click compute to obtain . The result shows dimensions, entrywise mapping, and (optionally) properties verified on your input.
- Toggle options for conjugate transpose, block transpose, and property checks like .
- Copy results directly into notes, LaTeX, or a programming array literal.
Examples
- Rectangular matrix: ⇒ .
- Vector conversion: (3×1) ⇒ (1×3).
- Product rule check: . Compute , so . Also .
- Symmetric matrix: ⇒ .
Formula Snippets Ready for Rendering
FAQs
What does transposing a matrix do?
It swaps rows and columns: entry moves to . The shape changes from m×n to n×m.
How do I transpose a row or column vector?
A column becomes a row and vice versa, using to denote the transposed vector.
Does the transpose change determinant or trace?
No. For square matrices, and .
Do A and AT have the same eigenvalues?
Yes. They share the same characteristic polynomial, so eigenvalues match (over appropriate fields).
What is the conjugate transpose?
For complex entries, take elementwise complex conjugate then transpose: .
Is transpose a linear operation?
Yes: .
How does transpose interact with multiplication?
It reverses order: . This is crucial in normal equations and orthogonality proofs.
Will transposing affect rank or norms?
Rank is preserved: . The Frobenius norm is also invariant.
Can I transpose block matrices?
Yes—transpose each block and swap their positions accordingly; the calculator can display the block‑transpose layout.
What if my matrix contains symbols?
Symbolic entries are transposed positionally. Algebraic relations are preserved for exact manipulations.