86 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA

has n derivatives so it makes sense to write Ly. Suppose Lyk = 0 for k = 1,2, · · · ,n.The Wronskian of these functions, yi is defined as

W (y1, · · · ,yn)(x)≡ det

y1 (x) · · · yn (x)y′1 (x) · · · y′n (x)

......

y(n−1)1 (x) · · · y(n−1)

n (x)

Show that for W (x) =W (y1, · · · ,yn)(x) to save space,

W ′ (x) = det

y1 (x) · · · yn (x)y′1 (x) · · · y′n (x)

......

y(n)1 (x) · · · y(n)n (x)

 .

Now use the differential equation, Ly = 0 which is satisfied by each of these func-tions, yi and properties of determinants presented above to verify that

W ′+an−1 (x)W = 0.

Give an explicit solution of this linear differential equation, Abel’s formula, and useyour answer to verify that the Wronskian of these solutions to the equation, Ly = 0either vanishes identically on (a,b) or never.

11. Two n× n matrices, A and B, are similar if B = S−1AS for some invertible n× nmatrix, S. Show that if two matrices are similar, they have the same characteristicpolynomials.

12. Suppose the characteristic polynomial of an n×n matrix, A is of the form

tn +an−1tn−1 + · · ·+a1t +a0

and that a0 ̸= 0. Find a formula A−1 in terms of powers of the matrix, A. Show thatA−1 exists if and only if a0 ̸= 0.

13. In constitutive modeling of the stress and strain tensors, one sometimes considerssums of the form ∑

∞k=0 akAk where A is a 3×3 matrix. Show using the Cayley Hamil-

ton theorem that if such a thing makes any sense, you can always obtain it as a finitesum having no more than n terms.

5.8 Shur’s TheoremEvery matrix is related to an upper triangular matrix in a particularly significant way. Thisis Shur’s theorem and it is the most important theorem in the spectral theory of matrices.

Lemma 5.8.1 Let{x1, · · · ,xn}