Question

What is the result of multiplying the Pauli-X matrix by the Pauli-Z matrix?

Accepted Answer

Multiplying the Pauli-X matrix by the Pauli-Z matrix results in the matrix equal to negative i times the Pauli-Y matrix, written as -iY. To understand this, first define the matrices: Pauli-X is [[0, 1], [1, 0]], Pauli-Y is [[0, -i], [i, 0]], and Pauli-Z is [[1, 0], [0, -1]], where i is the imaginary unit representing the square root of negative one. Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second. When you multiply X by Z, you perform the calculation on the first row of X and the first column of Z to get zero, the first row of X and the second column of Z to get zero, the second row of X and the first column of Z to get one, and the second row of X and the second column of Z to get negative one. This results in the matrix [[0, -1], [1, 0]]. Comparing this to the Pauli-Y matrix [[0, -i], [i, 0]], you can see that multiplying Y by negative i involves multiplying every element by negative i. Since negative i times i equals one, and negative i times negative i equals negative one, applying negative i to the elements of Y produces exactly the matrix [[0, -1], [1, 0]]. Therefore, the product of X and Z is equal to -iY, which is consistent with the fundamental commutation relation of Pauli matrices where the product of any two distinct Pauli matrices equals i times the third Pauli matrix in cyclic order, with the non-cyclic order resulting in a negative sign.

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What is the result of multiplying the Pauli-X matrix by the Pauli-Z matrix?