FAQ    

 

 

What is Inner product and outer product ? how they differ ?

The concepts of inner product and outer product are fundamental in mathematics, particularly in the fields of linear algebra and vector analysis. Each of these products provides different ways to combine vectors and has distinct applications and geometric interpretations.

Inner Product

  • Also known as: Dot product (for real-valued vectors), scalar product.
  • Operation: Takes two vectors of the same dimension and produces a single scalar value.
  • Geometric interpretation: Related to the angle between vectors and projections. A zero inner product implies orthogonality (perpendicularity).
  • Calculation (example): For vectors u = [1, 3, -2] and v = [4, 0, 1], their inner product is calculated as follows: 
    u ⋅ v = (1*4) + (3*0) + (-2*1) = 2
  • Notation: <u, v> or u ⋅ v

Outer Product

  • Operation: Takes two vectors (not necessarily of the same dimension) and produces a matrix.
  • Geometric Interpretation: Related to the area of a parallelogram formed by the vectors (or higher dimensional analogs).
  • Calculation (example): For vectors u = [1, 3, -2] and v = [4, 0], their outer product is: 
    [ 1  3  -2 ] 
[ 4  0  ]  =  [  4  0  -8  ]
               [ 12  0 -24 ]
               [ -8  0  16  ]
  • Notation: u ⊗ v

Key Differences

Feature Inner Product Outer Product
Input Two vectors of the same dimension Two vectors (can have different dimensions)
Output Scalar Matrix
Interpretation Angle, projection, orthogonality Area (parallelogram), scaling of dimensions
Applications Geometry, physics, machine learning concepts Covariance matrices, tensor products, some neural network architectures

Applications and Further Insights

  • Inner Product:
    • Physics: Calculating the work done by a force along a displacement.
    • Data Science: Used as a similarity measure in models like cosine similarity for text analysis.
    • Computer Graphics: Determines how much light (from a certain direction) will illuminate a surface.
  • Outer Product:
    • Statistics: Used to create covariance matrices which capture the variance and covariance of data sets.
    • Quantum Mechanics: Forms part of the foundation for describing states and operations in quantum theory.
    • Neural Networks: Employed in constructing certain types of network layers that transform input data through linear combinations based on the weights (outer products).