Multiplication Of Vectors with a Vector As An Element

[Fribbles]

Hi,

If I have a vector c = [ a , 1 ]^T where the element a is a vector. If I multiply c by its transpose:

c^Tc

is this defined? How do I calculate the a*a? Matrix multiplication rules would say that a * a is undefined because it is a nx1 matrix multiplied by a nx1 matrix. Or is the convention that it is the dot product?

Thank you in advance for your help!

 

[HallsofIvy]

In order to write vectors like that, you will need to think of the original vector, v, as a column matrix, say, v= \begin{bmatrix} x\\ y\end{bmatrix} so that its transpose is a row matrix: v*= \begin{bmatrix}x & y\end{bmatrix}. Then the product, v*u, is the matrix product \begin{bmatrix} x & y\end{bmatrix}\begin{bmatrix}a \\ b \end{bmatrix}= ax+ by.

Of course, that is only notation. If you want to write the vector v as a row, fine. Then its transpose is a column and you have to write the product in the other order. Or you can go with the more abstract concept- given a vector in a vector space, V, there exist an isomorphism from V to its dual, V*, the set of linear functions from V to the underlying field. In that case "v*u" is the linear function corresponding to v applied the vector u.

 

[sachav]

Are you sure this is a well-defined vector, though?
The matrix notation, \vec{u} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix} is defined such that x_1, \dots, x_n are the components of your vector in a given basis of your vector space, so they must be scalars.
That's why I don't think having a vector there makes much sense.

 

[Fribbles]

Thank you for the replies...so then does, a, have to be a column vector?

 

[genericusrnme]

when you take the dot product of two vectors a.b what you're really doing is the transpose of a and then doing matrix multiplication

so a.b === a^t b

if a is 1xn and b is nx1 then a^t is 1xn and so the matrix multiplication is defined (and gives you back a 1x1 matrix aka a real number)