Show that a bilinear form is an inner product

• CasinelliG
In summary: Every symmetric bilinear form can be diagonalized by succesively completing the square(try it!), until you´ve found a new basis for your vector space where the bilinear form is "diagonal", i.e. of the form x1y1 + x2y2 + ... + xkyk. If k is smaller than dimension of your space, you can easily see the bilinear form is degenerate(by example). In this case, it´s associated matrix is not invertible.
CasinelliG
Hi, I have a bilinear form defined as g : ℝnxℝn->ℝ by g(v,w) = v1w1 + v2w2 + ... + vn-1wn-1 - vnwn

I have to show that g is an inner product, so I checked that is bilinear and symmetric, but how to show that it's nondegenerate too?

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The map you have defined is not an inner product. Let $v = (0,0,\dots,0,1)$ and notice that $g(v,v)=-1 < 0$. Did you mean to write $g(v,w)=v^1w^1 + \dots + v^nw^n$?

Some physics texts define inner product as nondegenerate symmetric bilinear form.

To show this bilinear form is nondegenerate, write down its matrix and show it´s invertible (= nondegenerate).

Alesak said:
Some physics texts define inner product as nondegenerate symmetric bilinear form.

To show this bilinear form is nondegenerate, write down its matrix and show it´s invertible (= nondegenerate).

Yes, that's exactly the case. Now the book also says that g is nondegenerate if g(v, w) = 0 for all w in V implies v = 0 (in which V is an arbitrary vector space of dimension n ≥ 1).

How does this correlate to the existence of the inverse matrix?

Doesn't the symmetric part imply that the associated matrix (a_ij):=(q(vi,vj)) is symmetric. Then you have a symmetric matrix.

Symmetric matrices are diagonalizable, and non-degenerate means none of the diagonal entries is zero.

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Take a nonzero v. Then there is some vi that is nonzero. Consider w:=(0,...,0,1,0,...,0) where the 1 is in the ith place. Then g(v,w) is nonzero in both cases.

jgens: An inner product g such that g(v,v)>0 for all nonzero v is a particular case of an inner product called a euclidean inner product.

CasinelliG said:
How does this correlate to the existence of the inverse matrix?

Bacle2 said it. It can be also said in this way: every symmetric bilinear form can be diagonalized by succesively completing the square(try it!), until you´ve found a new basis for your vector space where the bilinear form is "diagonal", i.e. of the form x1y1 + x2y2 + ... + xkyk. If k is smaller than dimension of your space, you can easily see the bilinear form is degenerate(by example). In this case, it´s associated matrix is not invertible.

1. What is a bilinear form?

A bilinear form is a mathematical function that takes two vectors as inputs and returns a scalar value. It is linear in both of its arguments, meaning that if one of the vectors is held constant, the function behaves like a linear transformation with respect to the other vector.

2. What is an inner product?

An inner product is a type of bilinear form that is defined on a vector space over the real or complex numbers. It takes two vectors as inputs and returns a scalar value, but it also has several additional properties, such as symmetry, linearity in the first argument, and positive definiteness.

3. How do you show that a bilinear form is an inner product?

To show that a bilinear form is an inner product, you must demonstrate that it satisfies all of the properties of an inner product. These include symmetry, linearity in the first argument, and positive definiteness. You can also use specific mathematical tests, such as the Cauchy-Schwarz inequality, to prove that a bilinear form is an inner product.

4. What is the significance of a bilinear form being an inner product?

A bilinear form being an inner product is significant because it allows us to define a notion of length and angle in a vector space. This is especially useful in applications such as physics and engineering, where we need to measure distances and angles between vectors.

5. Can all bilinear forms be considered inner products?

No, not all bilinear forms can be considered inner products. A bilinear form must satisfy certain properties, such as symmetry and positive definiteness, in order to be considered an inner product. If a bilinear form does not satisfy these properties, it cannot be considered an inner product.

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