The Orthogonality of Vectors and Matrices

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SUMMARY

The discussion centers on the relationship between orthogonal vectors and orthogonal matrices. It is established that for three vectors \(v_1=(x_1,y_1,z_1)\), \(v_2=(x_2,y_2,z_2)\), and \(v_3=(x_3,y_3,z_3)\) to be orthonormal, they must satisfy the condition \(v_i \cdot v_j = \delta_{ij}\), where \(\delta_{ij}\) is the Kronecker delta. An orthogonal matrix has orthonormal columns, and its inverse is equal to its transpose.

PREREQUISITES
  • Understanding of vector operations, specifically dot products.
  • Familiarity with the concept of orthogonality in linear algebra.
  • Knowledge of matrices and their properties, particularly orthogonal matrices.
  • Basic understanding of the Kronecker delta function.
NEXT STEPS
  • Study the properties of orthogonal matrices and their applications in linear transformations.
  • Learn how to compute the dot product of vectors to verify orthonormality.
  • Explore the implications of orthonormal vectors in the context of vector spaces.
  • Investigate the use of the Kronecker delta in various mathematical proofs and applications.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as computer scientists and engineers working with transformations and vector spaces.

Petrus
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Hello MHB,
I wounder if I did understand correct, If we got 3 vector and they all are orthogonala, $$v_1=(x_1,y_1,z_1)$$,$$v_2=(x_2,y_2,z_2)$$,$$v_3=(x_3,y_3,z_3)$$ does that also mean that the matrix orthogonal so the invrese for the matrix is transport?

Regards,
$$|\pi\rangle$$
 
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Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.
 
Ackbach said:
Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.
How can I check if a columns are orthonormal? If I got it correct if $$v_1,v_2,v_3$$ shall be orthonormal that means that $$v_1*v_2*v_3=0$$

Regards,
$$|\pi\rangle$$
 
Check that $v_{i} \cdot v_{j}=\delta_{ij}$. Here
$$\delta_{ij}=\begin{cases}0,\quad &i \not=j \\ 1,\quad &i=j \end{cases}$$
is the Kronecker delta.
 

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