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I've been trying to study linear algebra online, and one term that confuses me is the word "normal". It seems like it has hundreds of meanings, and I'd like to clarify which purpose it serves in different contexts, and whether or not they are related.

For example, "normal vectors" can designate vectors that are normal to a subspace in the sense of being perpendicular or orthogonal, but I've also seen it being used to indicate vectors of unit length ("normalizing" a vector).

Since I've seen the term "orthonormal" vectors/basis being used to describe vectors that are orthogonal and of unit length, I don't see the purpose of using the term "normal" instead of orthogonal. I've read that there is a slight technical difference when going into higher mathematics, but I still don't understand it.

Then there is the expression "normal equations" that minimize the sum of the squares of errors between the original equations and those obtained from linear regression.

When I searched the query "normal in linear algebra", it then talked about "normal matrices" as well as matrices in "Jordan normal form".

I'd like to know the different meanings the term "normal" has and the purpose it serves in different contexts; whether or not its use is necessary, or whether mathematicians are simply being lazy again. Finally, are some of the definitions interconnected and chosen for a reason?

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    The word "normal" has two common uses - "normal" often means "perpendicular" or "orthogonal", but "normalized" often means "scaled to unit length" or having a magnitude of $1$. – K.defaoite Mar 19 '23 at 15:08
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    Yes, the word may have dozens of mathematical meanings (probably not hundreds). You have listed some of them. You tell which is meant by the context in which it appears. I agree that it's a little confusing at the start. I don't think it's an instance of "bad terminology again." Voting to close. – Ethan Bolker Mar 19 '23 at 15:17
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    I’m voting to close this question because not even the best terminology could replace a precise definition. – Kurt G. Mar 19 '23 at 15:51
  • The answers to an earlier and similar question of mine may be helpful. – user1551 Mar 19 '23 at 17:27

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This is a great question! According to Linear Algebra Done Right (3rd Edition) by Sheldon Axler, the term "normal" used in the context of operators on inner product spaces means the following:

Definition: normal - "An operator on an inner product space is called normal if it commutes with its adjoint"

If T is an operator on an inner product space, V, then its adjoint is given by T*. T is normal if the commutator of T and T* is zero, i.e. [T, T*] = 0.

Axler also discusses the "norm" of a vector which he defines as the square root of the inner product of a vector with itself. In the context of R^2 and R^3, the norm provides the length of the vector under consideration, however this is obviously not the general interpretation of the norm since one can also define the norm for R^n and other inner product spaces!

By defining the norm, we can address your question about orthonormal vectors. Axler provides the following definition:

Definition: orthonormal - "A list of vectors is called orthonormal if each vector in the list has norm = 1 and is orthogonal to all the other vectors in the list."

What this means is that orthogonality and normality of vectors are two different properties. Two vectors are orthogonal if their inner product is zero: <v,w> = 0. A vector is normal if its inner product with itself is one: <v,v> = 1. One can imagine a list of vectors that are all orthogonal to each other but are not normal in the sense that their inner product with themselves (their norm) is not equal to 1.

There are plenty of other ways "normal" is used in mathematics, but for linear algebra these are some of the most important uses. Thank you for your question!

Caleb Robinson
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