What does a matrix with rank 1 look like? Watch this video and find out! Featuring the outer product, a close companion to the dot product Check out my Matrix Algebra playlist: • Matrix Algebra Subscribe to my channel: / @drpeyam
Splendid! BTW, there's another meaning of "rank" of a matrix/tensor; namely, the number of indices it takes to specify an element of it; IOW, the dimension of the array of its elements. So that a scalar has rank=0, a vector has rank=1, a matrix has rank=2, etc. There ought to be some way of distinguishing these two very different terms. Maybe, "image rank" (because it tells the dim. of its image space) and "tensor rank" ? Also, another term for "outer product," is "tensor product," symbolically: ֿu ֿvᵀ = ֿu ⨂ ֿv Fred
@@weinihao3632 Yes, that could work. But there's still the other half of the problem - when "rank" comes up, meant as image space dimension, and the context is unclear, how do we tell - is there a good term for that, which is unambiguous? Fred
i thought the inner/outer product names indicated if the product payed attention to the parelle or normal parts of the vectors. the inner product is a product of u with the part of v inside u's span and the outer product is a product of u with the part of v outside u's span. does this outer product relate to orthogonality at all?
That's in geometric álgebra. The inner product and the "exterior" product, not "outer", when applied using the geometric product to a vector, descompose that vector in parallel and perpendicular components of another.
@@rajinfootonchuriquen i have seen some variations of geometric algebra use outer and exterior interchangeably. i had thought the convention for the name was far reaching than that. seeing as it actually gives a logic to how a product can inside something or outside something.
@@jonasdaverio9369 There is indeed an outer product in diff. geom., but it isn't this one. I believe that one is also known as a wedge product, dx⋀dy ; but it's been a long time, and I'm not quite sure about that. Fred
I think the outer product is just the tensor product between two vectors. Maybe dx wedge dy is simply the tensor product between two covector field (which would be a field of bilinear form). Can someone confirm?
@@drpeyam Calculate style The content of an image is represented by the values of the intermediate feature maps. It turns out, the style of an image can be described by the means and correlations across the different feature maps. Calculate a Gram matrix that includes this information by taking the outer product of the feature vector with itself at each location, and averaging that outer product over all locations
I think of the matrix multiplication of a 3x1 vector say the Cartesian x unit vector with the transposed Cartesian y unit vector which after transposing is a 1x3…. So 3x1 x 1x3 = 3x3 which is an outer product I believe… I think the bivector for the x-y plane?
