- #1
fisico30
- 374
- 0
Hello Forum,
When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors:
x= a1 v1 + a2 v2 + a3 v3+ ...
Each coefficient a_i is the dot product between x and each basis vector v_i.
If the vector x is not a row (or column vector), but an array (like an image) the equation is still the same: the a_i are single numbers, while the basis vectors v_i become arrays. The array x becomes the weighted sum of multiple basis arrays.
But how does the dot product between two matrices, x and v1 for example, both of size NxN, give a single number, the coefficient a1? The dimension does not seem to allow this matrix product to output a 1x1 vector (the single coefficient), does it?
a1=<x^T, v1>= dot product between the transpose of x and array v1...
thanks
fisico30
When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors:
x= a1 v1 + a2 v2 + a3 v3+ ...
Each coefficient a_i is the dot product between x and each basis vector v_i.
If the vector x is not a row (or column vector), but an array (like an image) the equation is still the same: the a_i are single numbers, while the basis vectors v_i become arrays. The array x becomes the weighted sum of multiple basis arrays.
But how does the dot product between two matrices, x and v1 for example, both of size NxN, give a single number, the coefficient a1? The dimension does not seem to allow this matrix product to output a 1x1 vector (the single coefficient), does it?
a1=<x^T, v1>= dot product between the transpose of x and array v1...
thanks
fisico30