Vector Spaces
Maths and statistics discussion, revision, exam and homework help.
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Vector SpacesA matrix
represents a linear transformation 
with respect to the basis 
where the matrix is 
.
If can be represented by a diagonal matrix 
, find and a basis of 
with respect to which represents . Also find non-singular matrix 
such that 
, where 
is the matrix representing with respect to 
.
If
and 
. How would I go about finding the new basis?
Last edited by JBKProductions; 16-04-2012 at 15:06. -
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Re: Vector SpacesAre the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix(Original post by nuodai)
The new basis is the set of columns of
. A matrix is diagonal if and only if it is represented w.r.t. a basis of eigenvectors.
would depend on how you write out the eigenvectors?
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Re: Vector SpacesThe answers are certainly not unique, you can choose any basis of eigenvectors you like. You can try and make it 'more unique' by requiring your basis vectors to be normalized and satisfy some kind of orientation relation -- in fact, in some cases (e.g. Hermitian matrices) you can even take your basis to be orthonormal -- but even then you're still free to change sign and so on.(Original post by JBKProductions)
Are the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix would depend on how you write out the eigenvectors?
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Re: Vector SpacesEven if an eigenvalue has multiplicity 1 then there is plenty of choice: Any non-zero vector in a 1-dimensional space is a basis.(Original post by JBKProductions)
Are the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix would depend on how you write out the eigenvectors?
