Hello, I need help with this problem

Let be a matrix, a and a such that .

Show:

1. is an invariant subspace relative to .

2. If is of column rank , the every eigenvalue of is an eigenvalue of .

Thanks in advance.(Bow)

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- Aug 9th 2010, 10:30 PMakolmanInvariant Subspace Problem
Hello, I need help with this problem

Let be a matrix, a and a such that .

Show:

1. is an invariant subspace relative to .

2. If is of column rank , the every eigenvalue of is an eigenvalue of .

Thanks in advance.(Bow) - Aug 20th 2010, 05:03 AMhalbardInvariant Subspace ProblemQuote:

1. is an invariant subspace relative to .

Let . Then for some .

Hence and so ; is therefore invariant under .

Quote:

2. If is of column rank , the every eigenvalue of is an eigenvalue of .

Now let be an eigenvalue of with non-zero eigenvector .

Let so that, by the remark above, is non-zero. Then

.

Since is non-zero, clearly is an eigenvalue of .