Square Matrix

• Square Matrix

 

 

A matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. 

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• Row Vector Transformation

 

$\mathbf{v}{R}^{\mathsf{T}}$

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• Why R need to be transposed instead of v?

 

- Transposing v would require additional adjustments to subsequent operations to revert back into a row vector. 

If you transpose 𝑣 , the result would initially be in a column format (i.e., $R({\mathbf{v}}^{\mathrm{\top }})$ ), and you'd need to transpose it back again to make it consistent with a row vector representation.

Using $\mathbf{v}{R}^{\mathsf{T}}$ avoids this extra step.

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For example,

 

Suppose $R=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ (a 90° rotation), and $\mathbf{v}=\left[\begin{array}{cc}x& y\end{array}\right]$ (a row vector).

1. If you transpose v, you would calculate:
$R({\mathbf{v}}^{\mathsf{T}})=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-y\\ x\end{array}\right]$
Then, to restore 𝑣 as a row vector, you'd need to transpose the result:
${\mathbf{v}}^{\mathrm{\prime }}=\left[\begin{array}{cc}-y& x\end{array}\right]$

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2. If you instead use $\mathbf{v}{R}^{\mathsf{T}}$, the calculation directly yields:
$\mathbf{v}{R}^{\mathsf{T}}=\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]=\left[\begin{array}{cc}-y& x\end{array}\right]$