Diagonalization

We show how to define a function of a square matrix using diagonalization procedure. This method is applicable only for such matrices, and is not suatable for defective matrices. Recall that a matrix A is called diagonalizable if there exists a nonsingular matrix S such that \( {\bf S}^{-1} {\bf A} {\bf S} = {\bf D} , \) a diagonal matrix. In other words, the matrix A is similar to a diagonal matrix. An \( n \times n \) square matrix is diagonalizable if and only if there exist n linearly independent eigenvectors, so geometrical multiplicities of each eigenvalue are the same as its algebraic multiplicities. Then the matrix S can be built from eigenvectors of A, column by column.

Let A be a square \( n \times n \) diagonalizable matrix, and let D be the corresponding diagonal matrix of its eigenvalues:

\[ {\bf D} = \begin{bmatrix} \lambda_1 & 0 & 0 & \cdots & 0 \\ 0&\lambda_2 & 0& \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0&0&0& \cdots & \lambda_n \end{bmatrix} , \]

where \( \lambda_1 , \lambda_2 , \ldots , \lambda_n \) are eigenvalues (that may be equal) of the matrix A.

Let \( {\bf x}_1 , {\bf x}_2 , \ldots , {\bf x}_n \) be linearly independent eigenvectors, corresponding to the eigenvalues \( \lambda_1 , \lambda_2 , \ldots , \lambda_n .\) We build the nonsingular matrix S from these eigenvectors (every column is an eigenvector):

\[ {\bf S} = \begin{bmatrix} {\bf x}_1 & {\bf x}_2 & {\bf x}_3 & \cdots & {\bf x}_n \end{bmatrix} . \]
For any reasonable (we do not specify this word, it is sufficient to be smooth) function defined on the spectrum (set of all eigenvalues) of the diagonalizable matrix A, we define the function of this matrix by the formula:
\[ f \left( {\bf A} \right) = {\bf S} f\left( {\bf D} \right) {\bf S}^{-1} , \qquad \mbox{where } \quad f\left( {\bf D} \right) = \begin{bmatrix} f(\lambda_1 ) & 0 & 0 & \cdots & 0 \\ 0 & f(\lambda_2 ) & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0&0&0& \cdots & f(\lambda_n ) \end{bmatrix} . \]