Solving Antisymmetric Matrix: Rx=zy & Ry=-zx

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In summary, an antisymmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. It is useful in solving systems of equations in the form of Rx=zy and Ry=-zx, and can be applied in various fields such as physics, engineering, and computer graphics. To solve an antisymmetric matrix, one can use the properties of matrix multiplication and isolate variables in given equations. Additionally, an antisymmetric matrix cannot have non-zero elements on the main diagonal, as they must be 0 due to the properties of matrix multiplication.
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Homework Statement


Hi, If I have an antisymmetric matrix R, how might I show that there are real 3D vectors x and y and some real number z such that
Rx=zy and Ry=-zx?
Thanks!

Homework Equations


Rx=zy and Ry=-zx

The Attempt at a Solution


I know that it is true intuitively because it is like a rotation matrix... but am not quite sure how to show it...
 
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Hi, problem solved! :)
 

1. What is an antisymmetric matrix?

An antisymmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In other words, for an nxn matrix A, aij = -aji if i < j.

2. What is the purpose of solving an antisymmetric matrix?

Solving an antisymmetric matrix can help us find solutions to systems of equations in the form of Rx=zy and Ry=-zx. This can be useful in various scientific fields, such as physics and engineering, where these types of equations appear frequently.

3. How do you solve an antisymmetric matrix?

To solve an antisymmetric matrix, we can use the properties of matrix multiplication and solve for the variables in the given equations. This involves isolating the variables on one side of the equation and using the properties of the matrix to manipulate the equation until we have a solution.

4. What are some applications of solving antisymmetric matrices?

Solving antisymmetric matrices can be applied in various fields of science and engineering, such as studying the motion of particles in physics, analyzing circuits in electrical engineering, and modeling fluid flow in fluid dynamics. It can also be used in computer graphics and image processing to perform transformations on images.

5. Can an antisymmetric matrix have non-zero elements on the main diagonal?

No, an antisymmetric matrix cannot have non-zero elements on the main diagonal. This is because the main diagonal elements are the negatives of themselves in an antisymmetric matrix, and any number multiplied by its negative is equal to 0. Therefore, the main diagonal elements in an antisymmetric matrix must be 0.

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