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Orthogonal and Unitary transformations; Rigid Motions Dentition: A matrix U such that U 1 is said to be unitary if its entries are complex, and orthogonal if they are real. Proposition: 1. U I U U
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How to fill out orthogonal and unitary transformatons
01
To fill out orthogonal transformations:
02
Start with a given matrix that represents a linear transformation.
03
Find the transpose of the given matrix.
04
Multiply the given matrix with its transpose.
05
If the result is the identity matrix, the transformation is orthogonal.
06
To fill out unitary transformations:
07
Start with a given matrix that represents a linear transformation.
08
Find the conjugate transpose of the given matrix.
09
Multiply the given matrix with its conjugate transpose.
10
If the result is the identity matrix, the transformation is unitary.
Who needs orthogonal and unitary transformatons?
01
Orthogonal transformations are useful in many areas such as computer graphics, robotics, signal processing, and physics.
02
They preserve angles, distances, and the dot product between vectors, making them suitable for applications involving rotations and reflections.
03
Unitary transformations, on the other hand, are commonly used in quantum mechanics and quantum computing.
04
They preserve the norm of vectors and the inner product between vectors, which is crucial in maintaining quantum information and ensuring the validity of quantum algorithms.
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