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Dot Product
The elements corresponding to the same row and column are multiplied together and the products are added such that, the result is a scalar.
Dot product of vectors a, b and c
Unlike matrix multiplication the result of dot product is not another vector or matrix, it is a scalar.
Dot product of vector a and b
Order of vectors does not matter for dot product, just the number of elements in both vectors should be equal.
The geometric formula of dot product is
Here |a| and |b| are magnitude of vector a and b and they are multiplied with cosine of angle between vectors
Dot product is also called inner product or scalar product.
Projection of Vector
Assuming that we have two vectors c and dsubtended by angle, phi(Ф).
Vector c with subscript-ed d represent projection of vector c on vector d
We can conclude from figure that the projection is equal to the horizontal component of vector c with respect to the angle phi(Ф).
This is called scalar projection.To find the vector projection of vector c on vector d we have to multiply scalar projection with unit vector d.
See also: What is 1/500 – Planning Map Scale 1/500, 1/2
Projections have wide use in linear algebra and machine learning (Support Vector Machine(SVM) is a machine learning algorithm, used for classification of data).
Hadamard Product (Element-wise Multiplication)
Hadamard product of two vectors is very similar to matrix addition, elements corresponding to the same row and columns of given vectors/matrices are multiplied together to form a new vector/matrix.
It is named after French Mathematician, Jacques Hadamard.
Hadamard product of vector g, H and m
The order of matrices/vectors to be multiplied should be the same and the resulting matrix will also be of the same order.
Matrix N is of the same order as input matrices (2×3)
Hadamard product is used in image compression techniques such as JPEG. It is also known as Schur product after German Mathematician, Issai Schur.
Hadamard Product is used in LSTM (Long Short-Term Memory) cells of Recurrent Neural Networks (RNNs).
See also: What is Check It Out – What does Check It Out mean
Additional Resources :
Dot Product
Vector Projection
Hadamard Product
Jacques Hadamard
Issai Schur
Use of Hadamard product in JPEG
LSTM
Read Part 15 : Orthogonality and four fundamental subspaces
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