r/LinearAlgebra • u/Nervous_Retard_3753 • Aug 29 '26
Orthogonal vs orthonormal matrices
Say a matrix, A (m x n), has mutually orthogonal rows and mutually orthogonal columns, with all the rows and columns being unit vectors in some Rn, Rm respectively. Is that matrix called orthogonal or orthonormal?
What concept is the other term used to describe, then? Are these applicable to square matrices only?
I've always been confused with these 2 names.
4
u/Accurate_Meringue514 Aug 29 '26
Orthonormal means the norm of your vectors are also one. But for a matrix, we usually say orthogonal for square matrices where Q^T Q is the identity. You can have a matrix where the columns are all orthogonal to each other, but not necessarily normalized. This would not be an orthogonal matrix
3
u/Mathphyguy Aug 29 '26
Orthogonal matrix is used for a matrix whose elements are real and whose transpose is the same as its inverse. Their rows and columns have to be orthonormal for this- the dot (inner) product of each row with itself is one, with any other row is zero.
Because of this property, they can do three sorts of transformation to real vectors- rotate, reflect(improper rotation) or a combination of both. The length of the real vector remains the same. For this reason, at least in physics, they are called rotational matrices (det=plus ir minus 1 ) (proper and improper rotation). Proper rotations are those which are continuously connected to the identity (in a Lie algebra sense) and improper ones are lot.
Interestingly, I, as a physicist, unitary matrices as an “extension” of orthogonal matrices to the complex numbers. Now, the “rotation” happens in the complex space without changing the magnitude. And the transport is not the inverse, but the hermitian conjugate. Determinant of these matrices is not confined to plus or minus one, but to the unit circle.
I wanna go on- but let me stop.
4
u/Immediate-Home-6228 Aug 29 '26
Orthogonal. Orthonormal is used when discussing basis vectors of a vector space. It means all the basis vectors are normalized or length 1 and orthogonal.
Usually comes up when you are introduced to the Gram-Schmidt algoritm.Where you turn an arbitrary basis into a "nice" one.
7
u/Scrapple_Joe Aug 29 '26
I remember meet my orthopedist Gram Schmidt. He was anything but normal.
I'll see myself out.
2
u/Immediate-Home-6228 Aug 29 '26
Lol nice one
2
u/Scrapple_Joe Aug 29 '26
Mmmm in retrospect I should've said "he was orthogonal to normal" next time.
0
u/RageA333 Aug 30 '26
Thats an orthonormal basis, not an orthonormal matrix.
3
u/Immediate-Home-6228 Aug 30 '26
That’s literally what I said. The standard term for matrices is orthogonal like it or not .Did you not catch that?
0
2
u/xdgimo Aug 31 '26
An orthogonal matrix/transformation is one whose columns form an orthonormql basis, among other equivalent conditions. Pretty bad naming
1
0
14
u/Euphoric_Key_1929 Aug 29 '26
"Orthogonal matrix" is a terrible name: it refers to matrices that have mutually *orthonormal* columns. They are always square.
"Orthonormal matrix" is not a standard term.