Maths - Bilinear Multiplication

We start by reviewing linearity and then extend this to bilineararity.

Linear Multiplication.

To represent linear space we only need two operations:

If the space is linear we have:

a*(v +u) = a*v + a*u

where:

In other words the two operations are distributive.

vector space

Bilinear Multiplication

To have bilinearity we need to include a multiplication where both operands are vectors.

bilinear multiplication

For a bilinear multiplication, if we keep one of the vectors constant and then the other vector varies in a linear way, as defined above.

For example, in the diagram above we take the product as the area formed from the two vectors. So if the area formed from V and U is V•U then if we multiply V by a scalar value 'a' we get the area a*V•U, so,

(a*V)•U = a*(V•U)

and similarily if we multiply U by b:

V•(b*U) = b*(V•U)

so combining these gives:

(a*V)•(b*U) = ab*(V•U)

An example of this type of multiplication is the vector dot product.

However this type of multiplication takes two vectors as input and produces a scalar value as an output. Is it possible to have a bilinear multiplication which takes two vectors as input and produces a vector as an output?

The requirement is:

(a V + a' V')×(b U + b' U') = ab(V×U) + ab'(V×U') +a'b(V'×U) +a'b'(V'×U')

this applies to the vector cross product.

The vector cross product only applies in 1,3 and 7 dimensions. However there are ways to generalise the vector cross product to other numbers of dimensions.

Quadratic Form

If the multiplication is also commutative then the bilinear form is a quadratic form as described on this page.


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

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This book contains more mathematically rigorous methods for picking than described above.

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