Proving Orthogonal Matrix with Identity Matrix and Non-Zero Column Vector a

In summary, to show that I - \frac{2}{| a |^{2}}aa^{T} is an orthogonal matrix, we need to use the inner product (dot product) for the non-zero column vector a. This is just a normalization factor and it is not the determinant. Additionally, it is important to remember that aa^{T} does not equal a^{T}.a, and matrix multiplication should be used instead. There is no need to find the determinant in this problem.
  • #1
sherlockjones
31
0
Assume that [tex] I [/tex] is the [tex] 3\times 3 [/tex] identity matrix and [tex] a [/tex] is a non-zero column vector with 3 components. Show that:
[tex] I - \frac{2}{| a |^{2}}aa^{T} [/tex] is an orthogonal matrix?My question is how can one take the determinant of [tex] a [/tex] if it is not a square matrix? Is there a flaw in this problem?

Thanks
 
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  • #2
I assume you are referring to the [tex] | a |^{2} [/tex] and I also assume that is the inner product (dot product) for the vector. It's just a normalization factor
 
  • #3
Yes. |a| is not a "determinant", it is the length of the vector a.
 
  • #4
Remember that [itex]aa^{T} [/itex] does NOT equal [itex]a^{T}.a[/itex], the scalar product. Use matrix multiplication. You don't need to find the determinant of anything either.
 

FAQ: Proving Orthogonal Matrix with Identity Matrix and Non-Zero Column Vector a

What is an orthogonal matrix?

An orthogonal matrix is a square matrix that has a special property where its columns and rows are all orthogonal to each other. This means that the dot product of any two columns or rows is equal to 0.

How is an orthogonal matrix related to the identity matrix?

An orthogonal matrix is closely related to the identity matrix because it is also a square matrix with 1s along the main diagonal and 0s everywhere else. However, an orthogonal matrix can have any real number as its elements, while the identity matrix only has 1s.

What is the significance of a non-zero column vector in proving an orthogonal matrix?

A non-zero column vector is important because it serves as a test vector to determine if the matrix is orthogonal. If the dot product of the test vector and any of the matrix's columns or rows is equal to 0, then the matrix is orthogonal.

How do you prove that a matrix is orthogonal using the identity matrix and a non-zero column vector?

To prove that a matrix is orthogonal, you need to multiply the matrix by its transpose. If the resulting product is equal to the identity matrix, then the original matrix is orthogonal. You can also test this by using a non-zero column vector and checking if the dot product with any of the matrix's columns or rows is equal to 0.

What are some real-world applications of orthogonal matrices?

Orthogonal matrices are commonly used in fields such as computer graphics, signal processing, and quantum mechanics. They can be used to rotate and reflect objects in 3D space, filter signals for noise reduction, and represent quantum states in quantum computing algorithms.

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