Eigenvalues and diagonalizability

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The discussion centers on the relationship between eigenvalues and diagonalizability for an n x n matrix A. It is established that a matrix can have eigenvalues even if its columns are linearly dependent, as nontrivial solutions exist in this case. Conversely, for a matrix to be diagonalizable, its columns must be linearly independent, although exceptions exist, such as the zero matrix, which is diagonalizable despite having dependent columns. The distinction between the null space and column space is crucial in understanding these concepts.

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  • Understanding of eigenvalues and eigenvectors
  • Knowledge of linear independence and dependence
  • Familiarity with matrix diagonalization
  • Concept of null space and column space
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  • Study the properties of eigenvalues in non-invertible matrices
  • Explore the conditions for diagonalizability of matrices
  • Learn about the implications of the null space and column space in linear algebra
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Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and anyone seeking to deepen their understanding of eigenvalues and diagonalization.

fk378
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This is a concept question..
I'm having trouble understanding why for an n x n matrix A, in order to have eigenvalues, it must have linearly dependent columns (so that a nontrivial solution exists), but for the same A, in order to be diagonalizable, the columns must be linearly INdependent.

The basis for the eigenspace of the former would be the null space, but for the latter, the basis would be the column space since no free variables exist.
 
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It may be difficult to understand because neither of those statements is true. If the columns (or rows) are linearly independent, then it has an INVERSE. A noninvertible matrix can have plenty of eigenvalues. Nor do the columns have to be independent for it to be diagonalizable. The zero matrix is perfectly diagonalizable.
 

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