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# Orthogonal matrices preserve angles and lengths

Showing that orthogonal matrices preserve angles and lengths. Created by Sal Khan.

## Want to join the conversation?

• Is C inverse, or C transpose, also an orthonormal matrix? Thanks!
• Straightforward from the definition: a matrix is orthogonal iff tps(A) = inv(A). Now, tps(tps(A)) = A and tps(inv(A)) = inv(tps(A)). This proves the claim. You can also prove that orthogonal matrices are closed under multiplication (the multiplication of two orthogonal matrices is also orthogonal): tps(AB) = tps(B)tps(A)=inv(B)inv(A)=inv(AB). Hope this helps :)
• does adding two nxn orthogonal matrices result in an nxn orthogonal matrix
• Good question,

To answer a question like this you should first try some examples. The best examples are easy examples. So let's try some 1x1 matrices. There are only two orthogonal matrices given by (1) and (-1) so lets try adding (1) + (1)=(2). (2) is not orthogonal so we have found a counterexample!.

In general you will see that adding to orthogonal matrices you will never get another since if each column is a unit vector the sum of two unit vectors cannot be a unit vector.
• You show cos(teta) = cos (teta_t), but i think it does not prove teta = teta_t, because teta_t also can be 360-teta, i.e., teta = -teta_t. Am i wrong or it does not matter?
• At , in order to prove that the length of a vector does not change when represented with respect to orthogonal basis, shouldn't we start by finding length of C^-1*X rather than C*X because X represented in basis B is equal to C^-1*X ?
• He's using C here to represent any transformation with matrices whose columns are orthonormal basis vectors. Its not important here that it can transform from some basis B to standard basis.

We know that the matrix C that transforms from an orthonormal non standard basis B to standard coordinates is orthonormal, because its column vectors are the vectors of B. But since C^-1 = C^t, we don't yet know if C^-1 is orthonormal. All we know is that its r o w vectors are an orthonormal set.
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• wait so x.y = x(t)y? i thought this was true iff x == y?
||cx|| = cx.cx = cx(t)cx using this logic i am.
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• At Sal mentions that orthonormal matrices "rotate [vectors] around." In the last video we saw an example of an orthonormal matrix reflecting vectors. Are there any other geometric transformations that an orthonormal matrix could produce?
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• No, those are the only two (though there can be combinations of rotation and reflection). Orthonormal matrices are special in that they preserve the relative orientation and size of vectors, so they will never scale a vector.
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• Previous videos are referred to, but I am unaware of the order of the videos. If I knew what video was prior and this video is confusing then I feel I may need to watch the preceding videos. I don't know the order.
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• You can access all the titles in order by just looking on the left, also by clicking on the titles at the upper left.
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• Since the angle between two vectors is preserved under reflections (when dealing with orthogonal transformation matrices, at least), how does one go about defining directed angles in R^n?
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• My guess is that in R^2 use delta(arctan(x2/x1)) between the two vectors. For R^3 and greater, which direction is positive?
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• Just because the cosine of the angle is preserved doesn't necessarily imply the angle is as well. What could be the case is that one of the vectors in question has now the opposite direction, and then, in turn, we have the angle being 180-theta, as by definition of angle between vectors we look at the angle in front of the (a-b) side. So should we instead be talking about the cosine being preserved rather than the angle??
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• How does one extend this proof (especially about the norms) to any finite-dimension inner product space?
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