Transpose of a Matrix

==2024-12-03

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Transpose yourself into something better?

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Theory

Transpose of a Matrix $A=[a_{ij}{]}_{m\times n}$, is defined as:

$$A^T=[a_{ji}{]}_{n\times m}$$

where, $1\le j\le m$ $1\le i\le n$

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Properties of Transpose

1. $(A^T{)}^T=A$
2. $(AB{)}^T=B^T{A}^T$
3. $(rA+sB{)}^T=r{A}^T+s{B}^T$ where $r,s\in \mathbb{R}$

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Symmetric Matrix

$A=A^T$

Skew-symmetric Matrix

$A=-A^T$

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PTR

1.Transpose of Matrices