■ dnn - r

Equation (11.14) is called the characteristic equation of matrix D. Since the determinant | D — rl\ will yield, upon Laplace expansion, an «th-degree polynomial in the variable r, (11.14) is in fact an «th-degree polynomial equation.

There will thus be a total of n roots, (/-, rn), each of which qualifies as a characteristic root. If D is symmetric, as is the case in the quadratic-form context, the characteristic roots will always turn out to be real numbers, but they can take either algebraic sign, or be zero.

Inasmuch as these values of r will all make the determinant | D - rl\ vanish, the substitution of any of these (say, r:) into the equation system (11.13') will produce a corresponding vector x\r_r. More accurately, the system being homogeneous. it will yield an infinite number of vectors corresponding to the root rr We shall, however, apply a process of normalization (to be explained below) and

* Characteristic roots are also known bv the alternative names of latent roots, or eigenvalues. Characteristic vectors arc also called eigenvectors.

select a particular member of that infinite set as the characteristic vector corresponding to r(; this vector will be denoted by vr With a total of n characteristic roots, there should be a total of n such corresponding characteristic vectors.

Example 5 Find the characteristic roots and vectors of the matrix substituting the given matrix for d in (11.14), we get the equation

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