Other than the matrix multiplication discussed earlier, vectors could be multiplied by two more methods : Dot product and Hadamard Product. Results obtained from both methods are different.Bạn đang xem: Dot product là gì
The elements corresponding to same row and column are multiplied together and the products are added such that, the result is a scalar.
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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 d, subtended by angle, phi(Ф).
Vector c with subscript-ed d represents 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).
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 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 same and the resulting matrix will also be of same order.
Matrix N is of 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.
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Additional Resources :
Use of Hadamard product in JPEG
Read Part 15 : Orthogonality and four fundamental subspaces
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I write about Maths, CS, ML, Statistics and Productivity | www.avnish.dev | linkedin.com/in/avnish-pal/Explaining the concepts of Linear Algebra and their application. View the complete series (in order) here : http://bit.ly/2UweRYgChuyên mục: Hỏi Đáp