Is it possible to have a diagonal matrix with all eigenvalues = zero ?

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A diagonal matrix with all eigenvalues equal to zero is the zero matrix, which has no linearly independent vectors. If at least one eigenvalue is zero, it implies that the matrix cannot have a full set of linearly independent eigenvectors. The zero matrix has only one eigenvalue, which is zero, and does not provide any linearly independent eigenvectors. The discussion clarifies that the original question was misphrased, focusing instead on the implications of having zero eigenvalues. Understanding these properties is crucial in linear algebra.
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Homework Statement




If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors?

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The Attempt at a Solution

 
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In response to the question in the subject, the zero matrix is diagonal and all its eigenvalues are zero.

In response to the question in the problem statement, if even one eigenvalue is zero, then by definition that means Ax = 0 for some nonzero x. Thus the columns of the matrix cannot be linearly independent.
 
The only eigenvalue of the zero matrix is 0. You can certainly find a set of linearly independent eigenvectors. ANY set of linearly independent vectors will do it. Is that all you are asking?
 
The question in the headline statement was a typo sorry.

Thx to answers

Meant to ask the question in the problem statement
 

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