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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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Hints for problems

Section (1.0)

2.    ρ(A + B) ≤ |||A + B|||2, where the matrix norm |||·|||2 is the spectral norm (see Section (5.6) of [HJ]). How are ρ(A) and |||A|||2 related when A is normal?

Section (1.2)

4.    Consider image and image.

15.    Show by example that they are not the same, but by (1.2.12) they both must be “angularly” the same. Thus, the only difference can be stretching or contracting along rays from the origin.

17.    The orthogonal complement of the intersection of the nullspaces of A and A* is the span of the union of the column spaces ...

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