More than 3 eigenvectors perpendicluar

[ss k]

all the eigenvectors of a matrix are perpendicular, ie. at right angles to each other, HOW?
I can imagine three eigenvectors as three perpendicular axes. How can be more than three axes are perpendicular with respect to each other?

 

[Nugatory]

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.

 

[ss k]

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.

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.

 

[HallsofIvy]

What do you understand by "vectors"? It is possible to have a vector space of any dimension (even infinite dimensional vector spaces are used). You seem to be thinking of vectors in three dimensional space. If that is what you mean, then, yes, you cannot have more than three mutually orthogonal vectors. But that is dealing with physics, not mathematics.

 

[FactChecker]

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.

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.

But if we can ignore the third axis, we can go to more dimensions and ignore more axis.

In four dimensional space, pick any two dimensions. They form a two dimensional space and the axes are at right angles. Ignore the other axis. Since the two you picked were arbitrary, all the axis are perpendicular to all the others.

This works for any number of higher dimensions.

 

[ss k]

Thanks all on PF. I think Factchecker replied exactly I was looking for.