18.6. THE MATRIX EXPONENTIAL, DIFFERENTIAL EQUATIONS ∗ 449

Lemma 18.6.1 Suppose Φ(t) is m×n and Ψ(t) is n× p and these are differentiable ma-trices. Then

(Φ(t)Ψ(t))′ = Φ′ (t)Ψ(t)+Φ(t)Ψ

′ (t)

Proof: By definition,

((Φ(t)Ψ(t))′

)i j =

((Φ(t)Ψ(t))i j

)′=

(∑k

Φ(t)ik Ψ(t)k j

)′= ∑

kΦ′ (t)ik Ψ(t)k j +∑

kΦ(t)ik Ψ

′ (t)k j

=(Φ′ (t)Ψ(t)

)i j +

(Φ(t)Ψ

′ (t))

i j

and so the conclusion follows. ■What do we mean when we say that for {Bn} a sequence of matrices

limn→∞

Bn = B?

We mean the obvious thing. The i jth entry of Bn converges to the i jth entry of B. One con-venient way to ensure that this happens is to give a measure of distance between matriceswhich will ensure that it happens.

Definition 18.6.2 For A,B matrices of the same size, define ∥A−B∥∞

to be

max{∣∣Ai j−Bi j

∣∣ , all i j}

Thus∥A∥

∞= max

{∣∣Ai j∣∣ , all i j

}To say that limn→∞ Bn = B is the same as saying that limn→∞ ∥Bn−B∥

∞= 0.

Here is a useful lemma.

Lemma 18.6.3 If A,Bn,B are p× p matrices and limn→∞ Bn = B, then

limn→∞

ABn = AB,

limn→∞

BnA = BA, (18.15)

Also∥AB∥

∞≤ p∥A∥

∞∥B∥

∞(18.16)∥∥∥Ak

∥∥∥∞

≤ pk−1 ∥A∥k∞

(18.17)

for any positive integer k and ∥∥∥∥∥ m

∑k=1

Ak

∥∥∥∥∥∞

≤m

∑k=1∥Ak∥∞

For t a scalar,∥tA∥

∞= |t|∥A∥

∞(18.18)

Also|Ax| ≤ √p∥A∥

∞|x| (18.19)

and∥A+B∥

∞≤ ∥A∥

∞+∥B∥

∞