The discussion revolves around the concept of eigenvectors and their perpendicularity in various dimensional spaces. Participants explore the mathematical implications of having more than three mutually perpendicular eigenvectors, particularly in higher-dimensional vector spaces.
Participants express differing views on the nature of perpendicularity in higher dimensions. While some agree on the mathematical validity of having more than three mutually perpendicular vectors, others emphasize the limitations of visualization in lower dimensions.
There is an implicit assumption that the discussion is grounded in mathematical definitions of vector spaces, which may not align with physical interpretations of dimensions.
This discussion may be of interest to those studying linear algebra, vector spaces, or anyone curious about the properties of eigenvectors in higher-dimensional mathematics.
Hi Nugatory,Nugatory said:In a two-dimensional space such as the surface of a sheet of paper you can only have two mutually perpendicular axes. In a three-dimensional space you can have three. You cannot visualize a four-dimensional space, but mathematically it's a perfectly sensible concept - and how many axes do you need there?
You just need a definition of "perpendicular" that's not tied to your ability to visualize 90-degree angles.
Consider the three dimensional case. Pick any two axis and ignore the third. Those two are in a 2 dimensional plane and the axes are at right angles.ss k said:Hi Nugatory,
Thanks usually there are n-vectors so n-axes and all are perpendicular with respect to each other. I don't want to plot them or see in some 3-d graphics programme but just wondering how they can even exist. Let's say I have 4 such axes (having values from -,0,+ and center orgin at 0) then axis 1,2,3 can be perpendicular with each other and how come the 4th axis will be perpendicular to any other axis.