Solving Orthogonal Matrix Problem

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In summary, to determine if a 2x2 matrix is orthogonal, we can use the condition alfa^2 + beta^2 = 1. This is the only condition needed as there are an infinite number of matrices that satisfy this condition.
  • #1
Niles
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[SOLVED] Ortogonal matrix

Homework Statement


I have a 2x2 matrix A:

[alfa beta]
[beta -alfa],

where alfa and beta are real parameters. I have to find out for which values of alfa and beta A is an orthogonal matrix.

The Attempt at a Solution


A matrix is orthogonal if it satisfies Q*Q^T = I.

So I will multiply A with A^T and equal it to I, and I get the condition alfa^2 + beta^2 = 1. Are there any other conditions I need?
 
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  • #2
ok.. so u have a matrix:

[tex]
\begin{bmatrix}
\alpha & \beta \\
\beta & -\alpha
\end{bmatrix}
[/tex]

on multiplying with it's transpose, you have:

[tex]
\begin{bmatrix}
\alpha^2 + \beta^2 & 0\\
0 & \beta^2 - \alpha^2
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
[/tex]

Look at the matrix now, equated with the identity matrix. You've taken one equation correctly. But, does the relation we have now provide you with another equation?

HINT: For two matrices to be equal, all their elements should be equal.
 
  • #3
Hmm, when multiplying with it's inverse (transpose), I get:[a^2+b^2 0 ]
[ 0 a^2+b^2].
 
Last edited:
  • #4
Yes, you were right the first time. Given only the information that A is a 2x2 orthogonal matrix you only have [itex]\alpha^2+ \beta^2= 1[/itex]. That's because there are an infinite number of such matrices, not just one. Any [itex]\alpha[/itex] and [itex]\beta[/itex] satisfying [itex]\alpha^2+ \beta^2= 1[/itex] will give you an orthogonal matrix.
 
  • #5
Cool, thanks.
 

FAQ: Solving Orthogonal Matrix Problem

What is an orthogonal matrix?

An orthogonal matrix is a square matrix in which all rows and columns are orthogonal to each other, meaning that they are perpendicular. This means that the dot product of any two rows (or columns) is equal to 0, and the dot product of a row (or column) with itself is equal to 1. In simpler terms, an orthogonal matrix is a matrix whose transpose is equal to its inverse.

What is the importance of solving orthogonal matrix problems?

Orthogonal matrix problems are important in many fields of mathematics and science, including linear algebra, signal processing, and quantum mechanics. They are useful in solving systems of linear equations, least squares problems, and in finding eigenvalues and eigenvectors. In addition, orthogonal matrices have properties that make them useful in computer graphics and other applications.

How do you solve an orthogonal matrix problem?

To solve an orthogonal matrix problem, you can use several methods such as the Gram-Schmidt process, the Householder transformation, or the QR decomposition. These methods involve manipulating the original matrix to transform it into an orthogonal matrix while preserving its important properties. Once the matrix is orthogonal, it becomes easier to solve problems involving it.

What are some real-world applications of solving orthogonal matrix problems?

Orthogonal matrix problems have numerous real-world applications. For example, they are used in image and signal processing to compress and filter data. They are also used in computer graphics to rotate and transform objects. In addition, orthogonal matrices are used in quantum mechanics to represent quantum states and operations.

Are there any challenges in solving orthogonal matrix problems?

Yes, there can be challenges in solving orthogonal matrix problems, especially when dealing with large matrices. One challenge is that some methods for solving these problems, such as the Gram-Schmidt process, can be numerically unstable, meaning that small errors in calculations can lead to significant errors in the final result. Another challenge is that these problems can become computationally intensive, requiring significant computing power and time to solve.

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