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captain
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what exactly are orthogonal matrices? can someone give me an example of how they would look like?
mathwonk said:The theorem classifying them is one of the few things in herstein's topics in algebra that is not in most other books.
captain said:if orthogonal matrices are for rotation them what unitary matrices for (or unitary groups)?
An orthogonal matrix is a square matrix in which the columns and rows are all orthogonal (perpendicular) to each other. This means that the dot product of any two columns or rows is equal to 0.
An orthogonal matrix is different from a regular matrix because it has the additional property of orthogonality. This means that the columns and rows are not only linearly independent, but also perpendicular to each other.
Some examples of orthogonal matrices include rotation matrices, reflection matrices, and the identity matrix. These matrices have special properties that make them useful in many mathematical and scientific applications.
Orthogonal matrices have many important applications in fields such as linear algebra, signal processing, and quantum mechanics. They are particularly useful for solving systems of equations, performing transformations, and preserving distances and angles.
To determine if a matrix is orthogonal, you can use the dot product test. Take the dot product of any two columns (or rows) and if the result is 0, the matrix is orthogonal. You can also check if the inverse of the matrix is equal to its transpose, which is another property of orthogonal matrices.