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Rotation Matrix

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Matrix is a Set of similar type of values. In mathematics, matrix is used in various applications like engineering mathematics, trigonometry, geometry, differential equation, etc. We can convert an object in 3D matrix into translate matrix or in the rotation matrix. Rotation matrix is used in the pair of matrix. Translate matrix (T) and rotation matrix (R) when combines then they make transform matrix (Tr).
We can understand it by this formula:
R-T=Tr
Here R, T and Tr are in the form of matrix.
It is very easy to find the rotation matrix from the transform matrix but finding the translate matrix from transform matrix, is very difficult.
When we talk about pair of rotation matrix and combining this matrix just remind that last column of transform matrix should not be included. But in Translation Matrix we include the last column.
Let us discuss the properties of rotation matrix, rotational matrix, and matrix rotation:
Property 1: Rotational is a kind of an orthogonal matrix which has the property as if we multiply this matrix with its transpose matrix we will get the identical matrix. We can show it as:
RRT= I
Here RT is the transpose of the matrix and I is an identical matrix.
Property 2: If we find the sum of element of any row or column then we will get the sum as 1.
Property 3: If we want to find the dot product of any two rows or column then it is always zero.
Property 4: In rotation matrix, rows of rotation matrix stand for coordinate which is in the original space.
Property 5: In rotation matrix, column stands for coordinates which is in rotated space.
We can use rotation matrix in 3D mathematics. If there is a matrix rotation, having three rows then row one will represent ‘right’, row two will represent upward direction and row three will represent the direction towards outside.

Rotation Matrix Proof

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A Rotation Matrix is a three dimensional matrix in space. It is a displacement of a rigid body along with a fixed Point. A body rotates about a fixed point that is known as a rotation. This rotation can be expressed in the form of matrix also; this is known as rotation matrix. We are here to discuss rotation matrix proof.
We can prove this with the help of Euler theorem which states that it is possible to find the Diameter of a Sphere, when it rotates around its center and direction of the movement is same as initial direction.
Let us see rotation matrix derivation:
To prove this let us assume a matrix of 3 * 3 we call it 'R', it moves from the coordinate 'x' to 'X', which is Rx =X. For 'n' number of rotations we can calculate the number of rotation as Rn = n.
According to the rotation matrix property the multiplication of a rotation matrix and its transpose makes a identity matrix.
R . RT = I,
If there rotation matrix is of order 3 * 3 then its identity matrix is also of 3 * 3 order. When we find the determinant from this relation we get the determinant as ±1.
Here determinant +1 denotes perfect rotation and negative determinant denotes improper rotation. Now we know that every rotation matrix has at least one vector as 'n' which means Rn = n. This requires (R – I) n = 0. If n = 1 then the det (R - I) = 0.
These two Relations are used to prove Euler theorem; that is det (-R) = - det(R) and det(R - 1) = 1.
With help of these rules we get det (R - ƛ I) = 0, for ƛ = 1.
For 'n' number of rotations we get (R – I) n = 0 which is equals to Rn = n.
This is the proof of Euler’s theorem for derivation of rotation matrix.