What is the "component projection" of vector a onto vector b, with the notation (a, b)?

Question

I stumbled across this notation while reading the article "A handwritten character recognition system using directional element feature and asymmetric Mahalanobis distance" (http://www.researchgate.net/publication/3192962_A_handwritten_character_recognition_system_using_directional_element_feature_and_asymmetric_Mahalanobis_distance), specifically the part about asymmetric Mahalanobis distance. I'm guessing it denotes scalar projection:

$(a,b) = a \cdot {b \over |b|}$

EDIT: This guess is probably wrong.

Answer 1

The actual context in the paper (at the beginning of section 5.2) is:

... Here, $u^i_j$ represents the component projection of $\mathbf v^i-\mu(i-1,2,\ldots, N)$ onto the vector $\phi_j(j=1,2,\ldots,N)$, i.e., $u^i_j=\Big(\mathbf v^i-\mathbf m,\phi_j\Big)$ ...

In this context, what I think makes sense is that $(\cdot,\cdot)$ is the standard inner product, such that $(\mathbf v^i-\mathbf m, \phi_j)$ is the component of $\mathbf v^i-\mathbf m$ in the direction of $\phi_j$ -- that it, the same thing that might otherwise be notated $(\mathbf v^i-\mathbf m)\cdot \phi_j$.

Thus $u^i_j$ for a particular $i$ and $j$ is just a number, whereas $(u^i_1,u^i_2,\ldots,u^i_n)$ is the representation of the vector $\mathbf v^i-\mathbf m$ in the $(\phi_j)_j$ basis.

(Either $\mu$ or $\mathbf m$ in the quote may be a typo for the other).

Answer 2

The projection of one vector onto (or on) another vector refers to the shadow of one vector along another. Just as we calculate the projection of one vector along x and y axis, similarly we can calculate projection of one vector along another. Its formula is: Acos@B A = mag. Of vector A @=angle between A and B B= direction of vector B;can be found using formula: B/|B| ; here B is vector B and |B| is mag. of B.

Hope it helps you. ; )