Eigenvalues and Eigenvectors uniquely define a matrix

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Eigenvalues and eigenvectors can define a matrix, but this is contingent upon the matrix being diagonalizable. If a matrix has complex eigenvalues, it may lack sufficient eigenvectors, complicating the reconstruction of the original matrix. The concepts of algebraic and geometric multiplicity of eigenvectors are crucial in understanding these limitations. A specific example, such as the 2x2 matrix [0, 1; 0, 0], illustrates these challenges. Thus, while eigenvalues and eigenvectors can uniquely define a matrix in certain cases, exceptions exist that must be considered.
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Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix M from a matrix of its eigenvectors as columns P and a diagonal matrix of the eigenvalues E through M=P E P^{\dagger}?
 
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i'm pretty sure the answer is yes
 
It is a bit more complicated. What you say is true if you can diagonalize the matrix. But take a matrix with complex eigenvalues and you are quickly missing eigenvectors. look up algebraic vs. geometric multiplicity of eigenvectors
 
Try it with 2 x 2 matrix [0, 1; 0, 0]
 

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