# Laub Matrix Analysis Theorem 9.15

From Alan J Laub, Matrix Analysis For Scientists and Engineers, 2004, p 79

Theorem 9.15. Let $$A \in \mathbb{C}^{n\times n}$$ have distinct eigenvalues $$\lambda_1, \ldots, \lambda_n$$ and let the corresponding right eigenvectors form a matrix $$X = [x_1, \ldots ,x_n]$$. Similarly, let $$Y = [y_1, \ldots ,y_n]$$ be the matrix of corresponding left eigenvectors. Furthermore, suppose the left and right eigenvectors have been normalized so that $$y_i^Hx_i=1, i \in \{1,\ldots,n\}$$. Finally, let $$\Lambda = \text{diag}(\lambda_1,\ldots,\lambda_n) \in \mathbb{R}^{n\times n}$$ ...

The theorem goes on, but I am wondering about the line where he lets $$\Lambda = \text{diag}(\lambda_1,\ldots,\lambda_n) \in \mathbb{R}^{n\times n}$$. In general, matrices over the complex numbers with distinct eigenvalues need not have real eigenvalues. How then can we assume that $$\Lambda \in \mathbb{R}^{n\times n}$$?

• Could very well be a typo, impossible to say without the complete statement of the theorem. – daw Oct 19 '18 at 6:03