Fundamental Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. This is an elegant bookkeeping tech nique and a very compact, efﬁcient way to express these formulas. As be fore, we state the deﬁnitions and results for a 2 × 2 system, but they gener alize immediately to n × n systems. We return to the system x� = A ( t ) x , (1) x = c 1 x1 ( t ) + c 2 x2 ( t ) , (2) with the general solution where x1 and x2 are two independent solutions to (1), and c1 and c2 are arbitrary constants. We form the matrix whose columns are the solutions x1 and x2 : � � � � x1 x1 x2 Φ(t) = = . x2 y1 y2 (3) Since the solutions are linearly independent, we called them a fundamen tal set of solutions, and therefore we call the matrix in (3) a fundamental matrix for the system (1). Writing the general solution using Φ(t). As a ﬁrst application of Φ(t), we can use it to write the general solution (2) efﬁciently. For according to (2), it is � � � � � �� x1 x2 x1 x2 c1 x = c1 + c2 = , y1 y2 y1 y2 c2 which becomes using the fundamental matrix � � c1 x = Φ(t) c where c = , (general solution to (1)). c2 (4) Note that the vector c must be written on the right, even though the c’s are usually written on the left when they are the coefﬁcients of the solutions xi . Fundamental Matrices OCW 18.03SC Solving the IVP using Φ(t). We can now write down the solution to the IVP x� = A ( t ) x , x ( t 0 ) = x0 . (5) Starting from the general solution (4), we have to choose the c so that the initial condition in (6) is satisﬁed. Substituting t0 into (5) gives us the matrix equation for c : Φ ( t 0 ) c = x0 . Since the determinant |Φ(t0 )| is the value at t0 of the Wronskian of x1 and x2 , it is non-zero since the two solutions are linearly independent (Theorem 3 in the note on the Wronskian). Therefore the inverse matrix exists and the matrix equation above can be solved for c: c = Φ ( t 0 ) − 1 x0 . Using the above value of c in (4), the solution to the IVP (1) can now be written x = Φ ( t ) Φ ( t 0 ) − 1 x0 . (6) Note that when the solution is written in this form, it’s “obvious” that x(t0 ) = x0 , i.e., that the initial condition in (5) is satisﬁed. An equation for fundamental matrices We have been saying “a” rather than “the” fundamental matrix since the system (1) doesn’t have a unique fundamental matrix: there are many ways to pick two independent solu tions of x� = A x to form the columns of Φ. It is therefore useful to have a way of recognizing a fundamental matrix when you see one. The following theorem is good for this; we’ll need it shortly. Theorem 1 Φ(t) is a fundamental matrix for the system (1) if its determi nant |Φ(t)| is non-zero and it satisﬁes the matrix equation Φ� = A Φ , (7) where Φ� means that each entry of Φ has been differentiated. Proof. Since |Φ| �≡ 0, its columns x1 and � x2 are � linearly independent, as we x1 saw in the previous note. Let Φ = . According to the rules for x2 matrix multiplication (7) becomes � � � � � � � x1 x1 Ax1 = A = . x2� x2 Ax2 2 Fundamental Matrices OCW 18.03SC which shows that x1� = A x1 and x2� = A x2 ; this last line says that x1 and x2 are solutions to the system (1). 3 �