Eigenvalues and Eigenvectors - Diagionalization
3 important questions on Eigenvalues and Eigenvectors - Diagionalization
Theorem 7
Let A be an n x n matrix whose distinct eigenvalues are lambda1, ..., lambdap.
a) For 1 smaller or equal than k smaller or equal to p, the dimension of the eigenspace for lambdak is
less than or equal to the multiplicity of the eigenvalue lambdak.
Theorem 7
Let A be an n x n matrix whose distinct eigenvalues are lambda1, ..., lambdap.
b) The matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n. This happens if and only if (i) & (ii), what are these?
(i) the characteristic polynomial factors completely into linear factors and
(ii) the dimension of the eigenspace for each lambdak equals the multiplicity of lambdak .
Theorem 7
Let A be an n x n matrix whose distinct eigenvalues are lambda1, ..., lambdap.
c) If A is diagonalizable and Bk is a basis for the eigenspace corresponding to lambdak
for each k, then what can be said about the total collection of vectors in the sets b1, ..., bp?
They form an eigenvector basis for Rn
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