By Charles A. Weibel
A portrait of the topic of homological algebra because it exists at the present time
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Additional resources for An Introduction to Homological Algebra
12 where Q is the product of real unitary matrices Qi above. When the block Pi is 2 × 2, its eigenvalues are a conjugate pair of eigenvalues of A and if it is 1 × 1 it is a real eigenvalue of A. Here is why this last claim is true λI1 − P1 · · · ∗ .. λI − T = . λIr − Pr 0 where Ik is the 2 × 2 identity matrix in the case that Pk is 2 × 2 and is the number 1 in the case where Pk is a 1 × 1 matrix. 5, det (λI − T ) = r ∏ k=1 det (λIk − Pk ) . Therefore, λ is an eigenvalue of T if and only if it is an eigenvalue of some Pk .
Proof: trace (AB) ≡ ( ∑ ∑ i k Aik Bki ) = ∑∑ k Bki Aik = trace (BA) i Therefore, ( ) ( ) trace (B) = trace S −1 AS = trace ASS −1 = trace (A) . 3 Let A be an n×n matrix. Then trace (A) equals the sum of the eigenvalues of A and det (A) equals the product of the eigenvalues of A. This is proved using Schur’s theorem and is in Problem 17 below. Another important property of the trace is in the following theorem. 13) z where A is a 3 × 3 symmetric matrix. In higher dimensions the idea is the same except you use a larger symmetric matrix in place of A.
If λ (s0 ) ∈ / Q, then from the above discussion there are λ (s) ∈ σ (A (s)) ∩ Q for s > s0 arbitrarily close to λ (s0 ) . Therefore, λ (s0 ) ∈ Q which shows that s0 > t because λ (t) is the only element of σ (A (t)) in D (λ (t) , δ) and λ (t) ∈ P. Now let sn ↑ s0 . Then λ (sn ) ∈ P for any λ (sn ) ∈ σ (A (sn )) ∩ D (λ (t) , δ) and also it follows from the above discussion that for some choice of sn → s0 , λ (sn ) → λ (s0 ) which contradicts P and Q separated and nonempty. Since P is nonempty, this shows Q = ∅.
An Introduction to Homological Algebra by Charles A. Weibel