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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; determinant < 4; (d) 3 (e) -1 < | A | < 1 for every eigenvalue >; no real eigenvalues; (f) each real eigenvalue has multiplicity one

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; determinant < 4; (d) 3 (e) -1 < | A | < 1 for every eigenvalue >; no real eigenvalues; (f) each real eigenvalue has multiplicity one

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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(d),(e),(f),(g)

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; 3 ≤ determinant < 4; (d) (e) -1 < | > | < 1 for every eigenvalue λ; (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; 3 ≤ determinant < 4; (d) (e) -1 < | > | < 1 for every eigenvalue λ; (f) no real eigenvalues; (g) each real eigenvalue has multiplicity one.
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