91Ó°ÊÓ

Let \(A\) be a matrix of order \(m \times n\), and let \(r\) and \(c\) denote the row and column rank of \(A\), respectively. Prove that \(r=c\). Hint: For convenience, assume that the first \(r\) rows of \(A\) are independent, with the remaining rows dependent on these first \(r\) rows, and assume the same for the first \(c\) columns of \(A\). Let \(\hat{A}\) denote the \(r \times n\) matrix obtained by deleting the last \(m-r\) rows of \(A\), and let \(\hat{r}\) and \(\hat{c}\) denote the row and column rank of \(\hat{A}\), respectively. Clearly \(\hat{r}=r .\) Also, the columns of \(\hat{A}\) are elements of \(\mathbf{C}^{r}\) which has dimension \(r\), and thus we must have \(\hat{c} \leq r .\) Show that \(\hat{c}=c\), thus proving that \(c \leq r\). The reverse inequality will follow by applying the same argument to \(A^{T}\), and taken together, these two inequalities imply \(r=c\).

Show that the infinite series $$ I+A+\frac{A^{2}}{2 !}+\frac{A^{3}}{3 !}+\cdots+\frac{A^{n}}{n !}+\cdots $$ converges for any square matrix \(A\), and denote the sum of the series by \(e^{A}\). (a) If \(A=P^{-1} B P\), show that \(e^{A}=P^{-1} e^{B} P\). (b) Let \(\lambda_{1}, \ldots, \lambda_{n}\) denote the eigenvalues of \(A\), repeated according to their multiplicity, and show that the eigenvalues of \(e^{A}\) are \(e^{\lambda_{1}}, \ldots, e^{\lambda_{n}}\).