### 7. Let ( A ) be a positive-definite ( n times n ) matrix. Let [ S(t)=sum_{m=0}^{infty} frac{(-1)^{m} A^{2 m} t^{2 m+1}}{(2 m+1) !} . ] (a) Show that this series of matrices converges uniformly for bounded ( t ) and its sum ( S(t) ) solves the problem ( S^{prime prime}(t)+A^{2} S(t)=0, S(0)= ) ( 0, S^{prime}(0)=I ), where ( I ) is the

7. Let ( A ) be a positive-definite ( n times n ) matrix. Let [ S(t)=sum_{m=0}^{infty} frac{(-1)^{m} A^{2 m} t^{2 m+1}}{(2 m+1) !} . ] (a) Show that this series of matrices converges uniformly for bounded ( t ) and its sum ( S(t) ) solves the problem ( S^{prime prime}(t)+A^{2} S(t)=0, S(0)= ) ( 0, S^{prime}(0)=I ), where ( I ) is the