Understanding 2x2 Matrices: Representation and Dot Product

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SUMMARY

The discussion clarifies the representation of a 2x2 matrix X using the equation X = a0 + σ · a, where σ represents the Pauli matrices and a is a vector of coefficients. This notation is shorthand for X = a0*I + σ1*a1 + σ2*a2 + σ3*a3, with I being the 2x2 identity matrix. The resulting matrix X is indeed a valid 2x2 matrix, combining scalar and matrix components through the dot product.

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  • Understanding of 2x2 matrices and their properties
  • Familiarity with Pauli matrices and their applications
  • Basic knowledge of matrix operations, including dot products
  • Concept of identity matrices in linear algebra
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Homework Statement


Suppose a 2x2 matrix X (not necessarily hermitian, nor unitary) is written as

X = a0 + sigma . a (the sigma . a is a dot product between sigma and a)

where a0 and a1, a2 and a3 are numbers.

How on Earth does X represent a matrix? it's a number added to another number (dot product).

Homework Equations

The Attempt at a Solution

 
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Josh Conneely said:

Homework Statement


Suppose a 2x2 matrix X (not necessarily hermitian, nor unitary) is written as

X = a0 + sigma . a (the sigma . a is a dot product between sigma and a)

where a0 and a1, a2 and a3 are numbers.

How on Earth does X represent a matrix? it's a number added to another number (dot product).

Homework Equations

The Attempt at a Solution


I suspect this is a question about Pauli matrices. That notation is shorthand for X=a0*I+sigma1*a1+sigma2*a2+sigma3*a3 where I is the 2x2 identity matrices and sigma1, sigma2 and sigma3 are 2x2 matrices collectively referred to as sigma. The result X is a 2x2 matrix.
 

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