Other than the matrix multiplication discussed earlier, vectors could be multiplied by two more methods : Dot hàng hóa & Hadamard Product. Results obtained from both methods are different.Dot Product

The elements corresponding to lớn same row & column are multiplied together và the products are added such that, the result is a scalar.

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Dot sản phẩm of vectors a, b & c

Unlượt thích matrix multiplication the result of dot product is not another vector or matrix, it is a scalar.



Dot product of vector ab

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 hàng hóa is


Here |a| và |b| are magnitude of vector ab và they are multiplied with cosine of angle between vectors

Dot hàng hóa is also called inner product or scalar product.

Projection of Vector

Assuming that we have sầu two vectors c & 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 lớn the horizontal component of vector c with respect to lớn 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.

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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 khổng lồ matrix addition, elements corresponding khổng lồ same row & columns of given vectors/matrices are multiplied together to khung a new vector/matrix.

It is named after French Mathematician, Jacques Hadamard.

Hadamard hàng hóa of vector g, hm

The order of matrices/vectors to lớn be multiplied should be same & the resulting matrix will also be of same order.

Matrix N is of same order as input matrices (2×3)

Hadamard hàng hóa is used in image compression techniques such as JPEG. It is also known as Schur sản phẩm after German Mathematician, Iskhông nên Schur.

Hadamard Product is used in LSTM (Long Short-Term Memory) cells of Recurrent Neural Networks (RNNs).

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Additional Resources :

Dot Product

Vector Projection

Hadamard Product

Jacques Hadamard

Iskhông đúng Schur

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 | | the concepts of Linear Algebra & their application. View the complete series (in order) here :

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