Inner product of rank 2 tensor and a vector

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SUMMARY

The discussion focuses on the inner product of a rank 2 tensor (dyad) and a vector, specifically examining the conditions under which this operation yields a vector of different magnitude but the same direction. The interaction is represented mathematically as A*v = η.v, where A is a dyad, v is a vector, and η is a scalar. The participant explores the implications of this relationship in a 2D context, analyzing how to set matrix values to satisfy the equations M x V = n x V.

PREREQUISITES
  • Understanding of rank 2 tensors and dyads
  • Familiarity with vector operations and inner products
  • Basic knowledge of matrix multiplication
  • Concept of scalar multiplication in linear algebra
NEXT STEPS
  • Study the properties of dyads in tensor algebra
  • Explore the geometric interpretation of inner products in vector spaces
  • Learn about eigenvalues and eigenvectors in relation to matrix transformations
  • Investigate conditions for scalar multiplication in linear transformations
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Mathematicians, physicists, and engineers interested in tensor analysis, linear algebra, and vector transformations.

abluphoton
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I been reading some material that lead me to understand that it takes an inner product of a dyad and a vector to obtain another vector at an angle to the initial one... cross product among two vectors would be an option only if we are willing to settle to a right angle.
After few days i countered a situation where i see an inner product of a vector and a dyad resulting in a vector of different magnitude but same direction as the earlier one. i mean to say
[ A*v = η.v ]
A= a random dyad
v= a vector
η= a scalar.

now the question is, what is the condition for such an interaction ?? what should be the property of such a tensor ?!
 
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look at a 2D version of a matrix and a vector say M x V = n x V

M x V = results in:

n*vx=m11*vx + m12*vy

and

n*vy = m21*vx + m22*vy

so what could you set the m values to make the two equations true?
 

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