A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two-dimensional. Is it possible that A is not diagonalizable? Justify your answer.
A matrix is diagonalizable if it is similar to a diagonal matrix. For an n x n diagonalizable matrix, the sum of the dimensions of its eigenspaces must equal n.
If a matrix is 4 x 4, the sum of the dimensions should be 4 for it to be diagonalizable. It is given that the matrix has 3 eigenvalues which means that to be diagonalizable, one of the eigenvalues is a double eigenvalue. A will not be diagonalizable if this double eigenvalue has only one independent eigenvector. However, it is given that one of the eigenspaces is two-dimensional - this will be from the double eigenvalue! The other two eigenvalues will contribute single eigenvectors. The these will provide the potential basis of eigenvectors of `R^4` .
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