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Each of the following properties defines a set of real n X n matrices. Find out which sets are dense, and which are open in the space L(R") of all linear opera- tors on R": (a) determinant = 0; (b) trace is rational; (c) entries are not integers;

Each of the following properties defines a set of real n X n matrices. Find out which sets are dense, and which are open in the space L(R") of all linear opera- tors on R": (a) determinant = 0; (b) trace is rational; (c) entries are not integers;

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Each of the following properties defines a set of real n × n matrices. Find out which sets are dense, and which are open in the space L(R") of all linear opera- tors on R": (a) determinant = 0; (b) trace is rational; (c) entries are not integers; (d) 3 determinant < 4; (e) 1<x< 1 for every eigenvalue X; (f) no real eigenvalues; (g) each real eigenvalue has multiplicity one.
Transcribed Image Text:Each of the following properties defines a set of real n × n matrices. Find out which sets are dense, and which are open in the space L(R") of all linear opera- tors on R": (a) determinant = 0; (b) trace is rational; (c) entries are not integers; (d) 3 determinant < 4; (e) 1<x< 1 for every eigenvalue X; (f) no real eigenvalues; (g) each real eigenvalue has multiplicity one.
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