Why is orthogonality important

I suspect this is what you had in the back of your mind for the first point. This is actually a very fundamental fact in geometry, regarding coordinate systems. I've once heard this described as the "second fundamental mistake in multivariable calculus" made by many students. This fact is immensely useful when dealing with infinite dimensional systems.

I also don't quite like your formulation of the third point. A vector is a vector is a vector. It is independent of the coordinate representation. So I'd expect that if two vectors are correlated assuming correlation is an intrinsic property , they better stay correlated without regard to choice of bases. What is more reasonable to say is that two vectors maybe uncorrelated in reality, but not obviously so when presented in one particular coordinate system, whereas the lack of correlation is immediately obvious when the vectors are written in another basis.

But this observation has rather little to do with orthogonality. It only has some relation to orthogonality if you define "correlation" by some inner product say, in some presentations of Quantum Mechanics. But then you are just saying that orthogonality of two vectors are not necessarily obvious, except when they are.

My personal philosophy is more one of practicality: the various properties of orthogonal bases won't make solving the problem harder. So unless you are in a situation where those properties don't make solving the problem easier, and some other basis does like what Ryan Budney described , there's no harm in prefering an orthogonal basis.

Furthermore, as Dick Palais observed above, one case where an orthogonal bases really falls out naturally at you is the case of the spectral theorem. The spectral theorem is, in some sense, the correct version of your point 3, that in certain situations, there is a set of basis vectors that are mathematically special.

And this set happens to always be orthogonal. Edit A little more about correlation. This is what I like to tell students when studying linear algebra. A vector is a vector. It is an object, not a bunch of numbers. When I hand you a rectangular box and ask you how tall the box is, the answer depends on which side is "up". This is similar to how you should think of the coordinate values of a vector inside a basis: it is obtained by a bunch of measurements.

Picking which side is "up" and measuring the height in that direction, however, in a non-orthogonal system, will require knowing all the basis vectors. See my earlier point. The point is that to quantitatively study science, and to perform numerical analysis, you can only work with numbers, not physical objects. So you have to work with measurements.

And in your case, the correlation you are speaking of is correlation between the measurements of I suppse different "properties" of some object.

And since what and how you measure depends on which basis you choose, the correlation between the data will also depend on which basis you choose. If you pick properties of an object that are correlated, then your data obtained from the measurements will also be correlated. The PCA you speak of is a way to disentangle that.

It may be difficult to determine whether two properties of an object is correlated. Maybe the presence of a correlation is what you want to detect. Orthogonality remains an important characteristic when establishing a measurement, design or analysis, or empirical characteristic. The assumption that the two variables or outcomes are uncorrelated remains an important element of statistical analysis as well as theoretical thinking.

The importance of orthogonality in research is an assumption that either is generated mathematically, assumed as part of the design, or established empirically. In each case, the importance Show page numbers Download PDF. Search form icon-arrow-top icon-arrow-top. Ooker 5 5 silver badges 21 21 bronze badges. Prateek Kulkarni Prateek Kulkarni 1 1 gold badge 6 6 silver badges 10 10 bronze badges. Add a comment. Active Oldest Votes. Furthermore, the above formula is very useful when dealing with projections onto subspaces.

Michael Albanese Michael Albanese With the equalities to not hold you mean that if we don't have orthonormal basis, the norm breaks, i. And this is the same for the scalar product too? And about the last paragraph, you say that the coordinate reps have the same length as the original vectors.

Which original vectors are you referring to? Is vector original if it's represented in an orthonormal basis? Rohit Gupta Rohit Gupta 1 2 2 bronze badges. The question is about what the additional assumption of orthogonality adds to linear independence.

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