Equivalent statements to matrix A is orthogonal

In summary, an orthogonal matrix is a square matrix with perpendicular rows and columns. It can be determined by checking if its transpose is equal to its inverse. Some properties of orthogonal matrices include eigenvalues with a magnitude of 1 and a determinant of 1 or -1. They are used in various fields such as geometry, linear algebra, and quantum computing. Any square matrix can be transformed into an orthogonal matrix through orthogonalization.
  • #1
CAVision
3
0
A is orthogonal if the A[itex]^{-1}[/itex] = A[itex]^{T}[/itex].

Thus, AA[itex]^{T}[/itex] = I.

However, is the statement A is orthogonal equivalent to A[itex]^{2}[/itex]=I.

I don't think the statements are equivalent, but it comes from a test. Thus, I'd hope the test is correct.
 
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  • #2
Multiply both sides of the A^2=I expression by A^-1 and see what you get.

From that expression you will be able to see the relationship required for both to be true.
 

1. What is an orthogonal matrix?

An orthogonal matrix is a square matrix in which all the rows and columns are orthogonal (perpendicular) to each other. This means that the dot product of any two rows or columns is equal to 0, resulting in a matrix that is "square" and "symmetric".

2. How do you determine if a matrix is orthogonal?

A matrix A is orthogonal if its transpose, AT, is equal to its inverse, A-1. This means that when you multiply A by its transpose, the result is the identity matrix, I. In other words, ATA = I.

3. What are some properties of orthogonal matrices?

Some properties of orthogonal matrices include:

  • All eigenvalues have a magnitude of 1, meaning they are either 1 or -1.
  • The determinant is either 1 or -1.
  • The inverse of an orthogonal matrix is equal to its transpose.
  • The columns and rows are orthonormal, meaning they are both orthogonal and normalized (have a magnitude of 1).

4. How are orthogonal matrices used in mathematics and science?

Orthogonal matrices are used in a variety of applications, including:

  • Rotations and reflections in geometry.
  • Transformations in linear algebra.
  • Signal processing and image compression.
  • Quantum mechanics and quantum computing.

5. Can any matrix be transformed into an orthogonal matrix?

Yes, any square matrix can be transformed into an orthogonal matrix through a process called orthogonalization. This involves finding a set of orthogonal vectors that span the same space as the original matrix, and then constructing an orthogonal matrix using these vectors.

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