Repeat Problem 30 for the Pauli spin matrix C in Problem 6....

Repeat Problem 30 for the Pauli spin matrix C in Problem 6. Hint: Show that if a matrix is diagonal, say $\mathrm{D}=\left(\begin{array}{cc}a & 0 \\ 0 & b\end{array}\right),$ then $f(\mathrm{D})=\left(\begin{array}{cc}f(a) & 0 \\ 0 & f(b)\end{array}\right).$

Key Concepts

Matrix Functions

Matrix functions extend the notion of classical functions defined on numbers to functions defined on matrices. When a matrix is diagonalizable, one can compute its function by applying the function to each of its eigenvalues; this is often accomplished via a power series representation of the function. The result is an operator that has been ‘transformed’ in the same way the scalar function transforms numbers.

Diagonal Matrices

Diagonal matrices are matrices in which all off-diagonal entries are zero. These matrices are particularly simple to handle under functions because the function acts separately on each diagonal element. Specifically, if D is a diagonal matrix, then applying a function f yields a diagonal matrix whose entries are f applied to the original diagonal entries. This simplicity is fundamental in the analysis and computation of functions of matrices.

Eigenvalue Decomposition

Eigenvalue decomposition is a technique whereby a matrix is expressed in terms of its eigenvalues and eigenvectors. For diagonalizable matrices, this decomposition allows one to represent the matrix as PDP?¹, where D is a diagonal matrix of eigenvalues. The process facilitates the computation of matrix functions because f(A) can then be written as P f(D) P?¹, where f(D) simply includes the function applied to each eigenvalue.

Pauli Matrices

Pauli matrices are a set of Hermitian and unitary matrices that are commonly used in quantum mechanics to describe spin operators. They have well-known eigenvalues and eigenvectors and serve as fundamental examples in the study of quantum spin systems. The properties of these matrices, including their diagonalizability and specific spectral characteristics, make them exemplary in illustrating the concept of applying functions to matrices.

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