Understanding 2x2 Matrices: Representation and Dot Product

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The discussion clarifies how a 2x2 matrix X can be represented using a combination of a scalar and a dot product involving Pauli matrices. It explains that the notation X = a0 + sigma . a actually denotes X as a linear combination of the identity matrix and the Pauli matrices, where a0 is a scalar and sigma represents the Pauli matrices. This formulation results in a valid 2x2 matrix. The key takeaway is that the expression combines scalars and matrices in a way that adheres to matrix algebra principles. Understanding this representation is essential for working with quantum mechanics and related fields.
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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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