the inner product
Givens two vectors (x,yin mathbb{R}^n), the quantity (x^ op y), sometimes called the inner product or dot product of the vectors, is a real number given by:
[x^ op y=egin{bmatrix}x_1 , x_2 ,cdots ,x_n end{bmatrix}egin{bmatrix}y_1 \ y_2 \ vdots \ y_nend{bmatrix}=sum_{i=1}^n x_iy_i
]
the inner products are reaully just special case of matrix multiplication.
the outer product
Given vectors (xinmathbb{R}^m,yin mathbb{R}^n)(not necessarily of the same size), (xy^ opinmathbb{R}^{m imes n}) is called the outer product of the vectors, It is a matrix whose entries are given by ((xy^ op)_{ij}=x_i y_i) ,i.e.,
[x y^ op=egin{bmatrix}x_1 \ x_2 \vdots \x_m end{bmatrix}egin{bmatrix}y_1, y_2, cdots, y_nend{bmatrix}=egin{bmatrix}x_1 y_1, x_1 y_2,cdots x_1 y_n\
x_2 y_1, x_2 y_2, cdots, x_2 y_n\ cdots, cdots, cdots ,cdots\
x_m y_1,x_m y_2, cdots, x_m y_nend{bmatrix}]