If *A* is an *n* by *n* real-valued matrix we say that *λ* is an *eigenvalue* of *A* with associated *eigenvector* **x** if

*A* **x** = *λ* **x**

Typically, a matrix *A* will have several different eigenvalue, eigenvector pairs. The standard method used to find these pairs is to note that

*A* **x** = *λ* **x**

if and only if

(*A* - *λ* *I*) **x** = 0

which will happen if and only if the matrix *A* - *λ* *I* is singular. One way to determine whether or not *A* - *λ* *I* is singular is to compute its determinant and check whether or not that determinant is 0. *det*(*A* - *λ* *I*) is an *n*^{th} degree polynomial in *λ* (called the characteristic polynomial of A), so the problem of finding eigenvalues boils down to the problem of finding roots of that polynomial.

There are a number of technical problems associated with finding eigenvalue, eigenvector pairs.

- Finding all the roots of a polynomial of high degree may be technically challenging.
- Not all roots of the characteristic polynomial may be real. Even though all of the entries of
*A*are real,*A*may still have some complex eigenvalues. Complex eigenvalues will also have complex eigenvectors associated with them. - Some roots may be repeated. In the case of repeated roots with multiplicity
*k*we may be able to find*k*linearly independent eigenvectors for that eigenvalue, or we may be able to find fewer than*k*associated eigenvectors.

Assuming for the moment that we can find a set of *n* distinct eigenvectors for some matrix *A* and also assuming that those eigenvectors form an orthonormal set, we can use those eigenvectors as a basis for the vector space *ℝ*^{n}. It turns out that such a basis is especially well suited to help us solve the matrix equation

*A* **x** = **b**

The solution method consists of expressing both **x** and **b** as linear combinations of eigenvectors.

*A* (*c*_{1} **u**_{1} + *c*_{2} **u**_{2} + ⋯ + *c*_{n} **u**_{n}) = *d*_{1} **u**_{1} + *d*_{2} **u**_{2} + ⋯ + *d*_{n} **u**_{n}

Since multiplication by *A* is a linear operation and the vectors **u**_{k} are all eigenvectors of *A* we have

*c*_{1} *λ*_{1} **u**_{1} + *c*_{2} *λ*_{2} **u**_{2} + ⋯ + *c*_{n} *λ*_{n} **u**_{n} = *d*_{1} **u**_{1} + *d*_{2} **u**_{2} + ⋯ + *d*_{n} **u**_{n}

Further, since we are assuming that the eigenvectors form a basis and are hence linearly independent, this equation has a solution if and only if

*c*_{k} *λ*_{k} = *d*_{k}

for all *k*. Since both *λ*_{k} and *d*_{k} are known, if none of the eigenvalues *λ*_{k} are 0 we can solve these equations for all of the *c*_{k} and then construct

**x** = *c*_{1} **u**_{1} + *c*_{2} **u**_{2} + ⋯ + *c*_{n} **u**_{n}

This method, which uses a basis of eigenvectors and their associated eigenvalues to solve a linear system is called the *spectral method*. (The list of eigenvalues of an operator is sometimes referred to as the *spectrum* of that operator, hence *spectral method*.)

The significance of this method is that it applies more broadly to problems involving linear operators. To solve

*f*(**x**) = **b**

for a linear operator *f* we try to find a basis of eigenvectors and associated eigenvalues:

*f*(**u**_{k}) = *λ*_{k} **u**_{k}

To solve *f*(**x**) = **b** we then proceed as above:

*f*(*c*_{1} **u**_{1} + *c*_{2} **u**_{2} + ⋯ + *c*_{n} **u**_{n}) = *d*_{1} **u**_{1} + *d*_{2} **u**_{2} + ⋯ + *d*_{n} **u**_{n}

*c*_{1} *λ*_{1} **u**_{1} + *c*_{2} *λ*_{2} **u**_{2} + ⋯ + *c*_{n} *λ*_{n} **u**_{n} = *d*_{1} **u**_{1} + *d*_{2} **u**_{2} + ⋯ + *d*_{n} **u**_{n}

This can be solved as above by solving the equations

*c*_{k} *λ*_{k} = *d*_{k}

for the unknowns *c*_{k}.

There is one very important special class of matrices for which the program outlined above works very nicely. These are the *n* by *n* *symmetric*, real-valued matrices. A matrix *A* is symmetric if it is equal to its own transpose. The key theorem that tells us that real-valued symmetric matrices are nice is the following.

**Theorem** If *A* is a real-valued, symmetric, *n* by *n* matrix, *A* has a complete set of associated real-valued eigenvectors. Further, eigenvectors corresponding to distinct eigenvalues are orthogonal to each other. Thus, it is possible to construct an orthonormal basis for *ℝ*^{n} consisting of eigenvectors of *A*.