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5.1–5.2. EIGENVALUES, EIGENVECTORS, EIGENSPACES,
CHARACTERISTIC EQUATION
If A is a square matrix, a scalar λ is called an eigenvalue of A if there
exists a column vector x ≠ 0 such that A x = λ x . If this is true, then each such
non-zero x is called an eigenvector for A associated with λ. There are other
standard phrases that are used for these concepts. An eigenvalue is also called a
characteristic value or a latent root. An eigenvector is also called a characteristic
vector. Since (for a specific λ) the set of all solutions of A x = λ x is a subspace,
we can call the set of all x, including the zero vector, which satisfy A x = λ x the
eigenspace associated with λ.
Notice that if λ = 0, then λ is an eigenvalue of A if and only if the
homogeneous system with A as coefficient matrix has at least one non-trivial
solution. Thus 0 is an eigenvalue of A if and only if A is singular (i.e., not
invertible).
Let’s consider some examples.
Example 1. Let A =
A x = λ x means that . So the eigenvalue
condition is x1 = λ x1 and 2x2 =λ x2.
If x1 ≠ 0, then it follows from the first requirement that we must have λ =
1 and then from the second requirement, 2x2 = x2, so that x2 = 0. So λ = 1 is an
eigenvalue of A and the associated eigenspace consists of all vectors in R2 of the
form (So is a basis for this eigenspace.) But if
x1 =0, then since x2 can’t also be zero (after all, x is assumed not to be the zero
vector.), it follows from the second requirement that λ = 2. So λ = 2 is the other
eigenvalue for A and the associated eigenspace consists of all vectors in R2 of the
form (And is a basis for this eigenspace.)
In the same way, one can show that for every n by n diagonal matrix, the
eigenvalues are the diagonal entries and for each k from 1 through n, the column
vector ek is an eigenvector associated with the kth entry of the matrix.
Example 2. Let A =
A x = λ x means that . So the
eigenvalue condition is x1 + x2 = λ x1 and 2x2 =λ x2. Now the second of the two
