Orthogonal matrices prove: T is orthogonal iff [T]_bb is an orthogonal matrix

In summary: T(y)]_B. Since we already know that [T(x)]_B . [T(y)]_B = [x]_B . [y]_B, this means that T is orthogonal, and we have proven the desired statement.
  • #1
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Homework Statement



Let B = {v1, ..., vn} be an arbitrary orthonormal basis of Rn, prove T is orthogonal iff [tex][T]_{BB}[/tex] is an orthogonal matrix.

Hint: If B is orhtogonal basis for Rn then, [tex]x.y = [x]_B . [y]_B[/tex]for all x, y in Rn.

3. The Attempt at a Solution

If [tex][T]_{BB}[/tex] is an orthogonal matrix then

1) [tex] ||[T(x)]_B|| = ||[x]_B|| [/tex]

2) [tex] [T(x)]_B . [T(y)]_B = [x]_B . [y]_B[/tex]

and since B is orthonormal,

[tex] ||[x]_B|| = ||x||[/tex]

[tex][x]_B . [y]_B = x.y[/tex]That's all I've got so far.. is this even right? How do I tie it into T being orthogonal?
 
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  • #2


Your approach is on the right track, but there are a few things that need to be clarified.

First, when you say "if [T]_{BB} is an orthogonal matrix", it's important to note that [T]_{BB} is the matrix representation of T with respect to the basis B. So when we say [T]_{BB} is orthogonal, we mean that it satisfies the property [T]_{BB}^T [T]_{BB} = I, where I is the identity matrix.

Secondly, in part 1) of your attempt, you have correctly stated that ||[T(x)]_B|| = ||[x]_B||, but it's important to note that this is true for all x in Rn, not just some arbitrary x. This is because ||[x]_B|| is the norm of the vector x with respect to the basis B, and ||[T(x)]_B|| is the norm of the vector T(x) with respect to the basis B. Since T is an orthogonal transformation, it preserves norms, so ||[T(x)]_B|| = ||x|| for all x in Rn.

In part 2) of your attempt, you have correctly stated that [T(x)]_B . [T(y)]_B = [x]_B . [y]_B, but again, this is true for all x and y in Rn, not just some arbitrary x and y. This is because [T(x)]_B and [T(y)]_B are the coordinate vectors of T(x) and T(y) with respect to the basis B, and [x]_B and [y]_B are the coordinate vectors of x and y with respect to the basis B. Since T is an orthogonal transformation, it preserves inner products, so [T(x)]_B . [T(y)]_B = [x]_B . [y]_B for all x and y in Rn.

Now, to tie this into T being orthogonal, we can use the fact that T is orthogonal if and only if it preserves inner products. This means that for any two vectors x and y in Rn, we have x.y = T(x).T(y). Using the hint provided, we can rewrite this as [x]_B . [y]_B = [T(x)]_B . [
 

1. What is an orthogonal matrix?

An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors, meaning they are perpendicular to each other and have a magnitude of 1. In other words, the dot product of any two columns or rows is equal to 0 and the magnitude of each column or row is 1.

2. What does it mean for a matrix to be orthogonal?

For a matrix to be orthogonal, it must have the property that its transpose is equal to its inverse. This means that multiplying an orthogonal matrix by its transpose will result in the identity matrix.

3. What does [T]_bb represent in this statement?

[T]_bb represents the matrix of linear transformation T with respect to the basis b. It is a change of basis matrix that maps the original basis to a new basis b.

4. How does the proof for this statement work?

The proof for this statement involves showing that if T is an orthogonal transformation, then [T]_bb is an orthogonal matrix. This is done by showing that the columns of [T]_bb are orthonormal vectors, which is a defining characteristic of an orthogonal matrix. Conversely, if [T]_bb is an orthogonal matrix, then T must be an orthogonal transformation, which can be proven by showing that the dot product of any two vectors under T is equal to the dot product of those same vectors under [T]_bb.

5. What are some applications of orthogonal matrices?

Orthogonal matrices are used in a variety of fields, including computer graphics, signal processing, and quantum mechanics. They are particularly useful for rotating, reflecting, and scaling objects in computer graphics and for transforming signals in signal processing. In quantum mechanics, orthogonal matrices are used to represent unitary transformations, which are essential for studying quantum systems.

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