The real dot product is just a special case of an inner product. In fact it's even positive definite, but general inner products need not be so. The modified dot product for complex spaces also has this positive definite property, and has the Hermitian-symmetric I mentioned above. Inner products are generalized by linear forms.

2017年7月14日 · The dot product is a particular example of an inner product, but there are many other inner products, such as the integral one defined above. And yes, orthogonality is defined in the same way, using whichever inner product you happen to be using.

The dot product is another name for an inner product on Rn R n and typically refers to the one you mentioned first. Inner products are more general but share many of the same properties with the usual dot product.

2020年12月15日 · In a space of uncountable dimension, the ∑ ∑ in the dot (outer) product becomes a single (double) integral. The cross product x × y x × y almost certainly refers to something exclusive to 3 3 dimensions, but sometimes it's something related but different.

np.inner is sometimes called a "vector product" between a higher and lower order tensor, particularly a tensor times a vector, and often leads to "tensor contraction". It includes matrix-vector multiplication. np.dot corresponds to a "tensor product", and includes the case mentioned at the bottom of the Wikipedia page.

2013年11月16日 · The inner product and dot product are two mathematical operations that are often confused with each other, but they have distinct differences. The main difference between the two is the type of vectors they operate on.

2018年8月2日 · I feel like there's some sort of convention with vector multiplication where the vector X X is treated as a column vector, and XTX X T X and XXT X X T are the inner and outer product of X X respectively. Is this true in all cases? Are there any tricks or mnemonics to help me keep track of the product of a string of such multiplications?

2012年4月14日 · The dot product is a particular example of an inner product. Or, equivalently, the notion of inner product generalizes the dot product. A physical example is that in Euclidean space, the dot product of two vectors is equal to the cosine of the angle between them.

2013年4月2日 · Hopefully this response will help others. The "double inner product" and "double dot product" are referring to the same thing- a double contraction over the last two indices of the first tensor and the first two indices of the second tensor. A double dot product between two tensors of orders m and n will result in a tensor of order (m+n-4).

2018年8月3日 · The inner product is something that respects this, while the dot product does not. In a similar manner, something of the form xTATAy x T A T A y also does not change under basis, which means that it is an inner product. However, this is also a …