Cross product of row and column vector

[Gear300]

For the two matrices A and B, (AB)_i,j = r_i . d_j ---- . refers to dot product ---- r_i is the ith row in A and d_j is the jth column in B.

Let us say that A and B are n x n system of column vectors. Then a row vector r_i of A would correlate to a component vector of the sum of the column vectors in A specified by the ith space. Wouldn't that imply that r_i . d_j would only be the product between the element d_j,i and the sum of the elements of r_i or am I just being too technical?

 

[Klockan3]

Is this a question about the common matrix multiplication?

 

[Gear300]

Is this a question about the common matrix multiplication?

Yup...the text said that if this dot product is 0 for all i =/= j and 1 for all i = j, then it carries geometric significance in that the rows of i in A and columns of j =/= i in B are orthogonal. I'm just trying to see this geometric significance.

In general, shouldn't the cross product of a 1 x n row vector in R¹ and an n x 1 column vector in Rⁿ space be the product between the sum of the elements in the row vector and the first element in the column vector?

 

[awkward]

Yup...the text said that if this cross product is 0 for all i =/= j and 1 for all i = j, then it carries geometric significance in that the rows of i in A and columns of j =/= i in B are orthogonal. I'm just trying to see this geometric significance.

In general, shouldn't the cross product of a 1 x n row vector in R¹ and an n x 1 column vector in Rⁿ space be the product between the sum of the elements in the row vector and the first element in the column vector?

I think you (or the book) must mean the dot product instead of the cross product. If the dot product of two vectors is zero then they are orthogonal.

The cross product of two vectors in R^n is not even defined if n > 3.

 

[Gear300]

I think you (or the book) must mean the dot product instead of the cross product. If the dot product of two vectors is zero then they are orthogonal.

The cross product of two vectors in R^n is not even defined if n > 3.

Heheh...a little typo there. I meant dot product.

What I'm having trouble understanding is what is being orthogonal to what; I figured that the row vector in A is considered a component of the vector sum of the matrix and has a direction specific to whatever value i is; but this vector is in R¹ space, whereas the column vector is in Rⁿ space with n > 1. Wouldn't this mean that the dot product with the row vector only applies to the element in row i for the column vector?

 

[Klockan3]

Heheh...a little typo there. I meant dot product.

What I'm having trouble understanding is what is being orthogonal to what; I figured that the row vector in A is considered a component of the vector sum of the matrix and has a direction specific to whatever value i is; but this vector is in R¹ space

The row vector is in Rⁿ space, since there are n indices for each row. You take the inner product between the whole row vector with the whole column vector. Row i of A times column j of B creates the element ij of the new matrix. The first element in row i times the first element in column j plus the second element in row i times the second element in column j etc.