A scalar \(\lambda\) is a proper value and a non-zero vector \(x\) is a proper vector of a linear transformation \(A\) if \(A x=\lambda x\) . Almost every combination of the adjectives proper, latent, characteristic, eigen, and secular, with the nouns root, number, and value, has been used in the literature for what we call a proper value. It is important to be aware of the order of choice in the definition; \(\lambda\) is a proper value of \(A\) if there exists a non-zero vector \(x\) for which \(A x=\lambda x\) , and a non-zero vector \(x\) is a proper vector of \(A\) if there exists a scalar \(\lambda\) for which \(A x=\lambda x\) .

Suppose that \(\lambda\) is a proper value of \(A\) ; let \(\mathcal{M}\) be the collection of all vectors \(x\) that are proper vectors of \(A\) belonging to this proper value, that is, for which \(A x=\lambda x\) . Since, by our definition, \(0\) is not a proper vector, \(\mathcal{M}\) does not contain \(0\) ; if, however, we enlarge \(\mathcal{M}\) by adjoining the origin to it, then \(\mathcal{M}\) becomes a subspace. We define the multiplicity of the proper value \(\lambda\) as the dimension of the subspace \(\mathcal{M}\) ; a simple proper value is one whose multiplicity is equal to \(1\) . By an obvious extension of this terminology, we may express the fact that a scalar \(\lambda\) is not a proper value of \(A\) at all by saying that \(\lambda\) is a proper value of multiplicity zero. The set of proper values of \(A\) is sometimes called the spectrum of \(A\) . Note that the spectrum of \(A\) is the same as the set of all scalars \(\lambda\) for which \(A-\lambda\) is not invertible.

If the vector space we are working with has dimension \(n\) , then the scalar \(0\) is a proper value of multiplicity \(n\) of the linear transformation \(0\) , and, similarly, the scalar \(1\) is a proper value of multiplicity \(n\) of the linear transformation \(1\) . Since \(A x=\lambda x\) if and only if \((A-\lambda) x=0\) , that is, if and only if \(x\) is in the null-space of \(A-\lambda\) , it follows that the multiplicity of \(\lambda\) as a proper value of \(A\) is the same as the nullity of the linear transformation \(A-\lambda\) . From this, in turn, we infer (see Section: Rank and nullity , Theorem 1) that the proper values of \(A\) , together with their associated multiplicities, are exactly the same as those of \(A^{\prime}\) .

We observe that if \(B\) is any invertible transformation, then \[B A B^{-1}-\lambda=B(A-\lambda) B^{-1},\] so that \((A-\lambda) x=0\) if and only if \((B A B^{-1}-\lambda) B x=0\) . This implies that all spectral concepts (for example, the spectrum and the multiplicities of the proper values) are invariant under the replacement of \(A\) by \(B A B^{-1}\) . We note also that if \(A x=\lambda x\) , then \begin{align} A^{2} x &= A(A x)\\ &= A(\lambda x)\\ &= \lambda(A x)\\ &= \lambda(\lambda x)\\ &= \lambda^{2} x. \end{align} More generally, if \(p\) is any polynomial, then \(p(A) x=p(\lambda) x\) , so that every proper vector of \(A\) , belonging to the proper value \(\lambda\) , is also a proper vector of \(p(A)\) , belonging to the proper value \(p(\lambda)\) . Hence if \(A\) satisfies any equation of the form \(p(A)=0\) , then \(p(\lambda)=0\) for every proper value \(\lambda\) of \(A\) .

Since a necessary and sufficient condition that \(A-\lambda\) have a non-trivial null-space is that it be singular, that is, that \(\operatorname{det}(A-\lambda)=0\) , it follows that \(\lambda\) is a proper value of \(A\) if and only if it is a characteristic root of \(A\) . This fact is the reason for the importance of determinants in linear algebra. The useful geometric concept is that of a proper value. From the geometry of the situation, however, it is impossible to prove that any proper values exist. By means of determinants we reduce the problem to an algebraic one; it turns out that proper values are the same as roots of a certain polynomial equation. No wonder now that it is hard to prove that proper values always exist: polynomial equations do not always have roots, and, correspondingly, there are easy examples of linear transformations with no proper values.