x for some scalar .
is called an eigenvalue of A if there is a nontrivial solution
x of Ax = x. In this case x is the eigenvector
corresponding to . (A "nontrivial" solution vector x is a vector that has at least
one nonzero entry.)
| eigenvector |
An eigenvector of an nxn matrix A is a nonzero vector x such that Ax =
x for some scalar .
|
| eigenvalue |
A scalar is called an eigenvalue of A if there is a nontrivial solution
x of Ax = x. In this case x is the eigenvector
corresponding to . (A "nontrivial" solution vector x is a vector that has at least
one nonzero entry.)
|
The symbol is the lowercase Greek letter Lambda and is frequently used to denote eigenvalues.
The equation Ax = x is called the eigenvalue-eigenvector equation.
Notice that x = 0 always satisfies this equation, which is why it is called the trivial solution. However, we never consider x = 0
to be an eigenvector. Reasons for this include the fact that 0 is linearly dependent with any set of vectors from Rn and also
that it would allow any scalar to be considered an eigenvalue.
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