## Nondegenerate Forms I

The notion of a positive *semi*definite form opens up the possibility that, in a sense, a vector may be “orthogonal to itself”. That is, if we let be the self-adjoint transformation corresponding to our (conjugate) symmetric form, we might have a nonzero vector such that . However, the vector need not be completely trivial as far as the form is concerned. There may be another vector so that .

Let us work out a very concrete example. For our vector space, we take with the standard basis, and we’ll write the ket vectors as columns, so:

Then we will write the bra vectors as rows — the transposes of ket vectors:

If we were working over a complex vector space we’d take conjugate transposes instead, of course. Now it will hopefully make the bra-ket and matrix connection clear if we note that the bra-ket pairing now becomes multiplication of the corresponding matrices. For example:

The bra-ket pairing is exactly the inner product we get by declaring our basis to be orthonormal.

Now let’s insert a transformation between the bra and ket to make a form. Specifically, we’ll use the one with the matrix . Then the basis vector is just such a one of these vectors “orthogonal” to itself (with respect to our new bilinear form). Indeed, we can calculate

However, this vector is not totally trivial with respect to the form . For we can calculate

Now, all this is prologue to a definition. We say that a form (symmetric or not) is “degenerate” if there is some non-zero ket vector so that for *every* bra vector we find

And, conversely, we say that a form is “nondegenerate” if for every ket vector there exists *some* bra vector so that

“We say that a form (symmetric or not) is “degenerate” if there is some ket vector ”

Do you want to say non-zero ket vector?

Comment by Johan Richter | July 19, 2009 |

Yes, sorry. I caught this in the next post, but didn’t here.

Comment by John Armstrong | July 19, 2009 |

[...] the orthogonal groups. This covers orthogonality with respect to general (nondegenerate) forms on an inner product space , the special case of orthogonality with respect to the underlying [...]

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