Tool for calculating a transition matrix (change of basis) based on a homothety or rotation in a vector space and coordinate change calculations.

Transition Matrix - dCode

Tag(s) : Matrix

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The transition matrix is the matrix allowing a calculation of change of coordinates according to a homothety or a rotation in a vector space.

From a transformation matrix $ P $ (also called base change of basis matrix), any vector $ v $ then becomes the vector $ v' $ in the new base by the computation (dot / multiplication">matrix product) $$ v' = P.v $$

__Example:__ $ \begin{bmatrix} v_1' \\ v_2' \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} . \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} $

**From a rotation angle** $ \alpha $ (trigonometric direction) and an axis, the rotation matrix is written as (rotation around the axis $ z $) $$ \begin {bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha \cos \alpha & 0 \\ 0 & 0 & 1 \ \end{bmatrix} $$

**From 2 vectors** (the original and the destination one), it is possible to generate an equation system to solve to find the values of $ \alpha $ and the axis.

From the value of the scaling factor $ k $ (homothety assumed to be uniform throughout the vector space of size $ n $), the passing matrix is given by the formula $ k.I_n $ (with $ I_n $ the identity matrix).

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*Transition Matrix* on dCode.fr [online website], retrieved on 2022-08-08,

transition,change,basis,matrix,vector,homothety,rotation,coordinate

https://www.dcode.fr/matrix-change-basis

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