Econ 407:

Notes on Characteristic Roots also known as latent roots (eigenvalues) of a matrix. One way to find the determinant of a matrix is to find the characteristic root(s) of the matrix and then multiplying them together. Carefully examine the example below. The detreminant of A is : (-6)(-6)-(3)(3)=27.

Now if we rewrite the matrix with the elments on the main diagnoal replaced by[ -6-lambda] for

(a11) and [-6-lambda[ for (a22), and find the detrmiants of this matrix, we will obtain a quadratic equation. If we solve this equation for lambda, the solutions are (-9) and (-3). These are known as the characteristic root of the matrix A. Now if you multiply these two roots we get 27-- which is the determiant of the orignal matrix.

> with(linalg):
A:=matrix([[-6,3],[3,-6]]);

> deta:=det(A);

> with(linalg):
A1:=matrix([[-6-lambda,3],[3,-6-lambda]]);

> deta1:=det(A1);

> solve(deta1,lambda);

> pro:=(-9)*(-3);

Characteristic roots have many applications, including finding the sign of a matrix. More specifically:

a. if all the roots are positive, then A is positive definite.

b. if all the roots are negative, then A is negative definite.

c. if all the roots are nonnegative and at least one of the roots =0, then A is positive semidefinite.

d. if all the roots are nonpositive and at least one of the roots =0, then A is negative semidefinite.

e.if some of the roots are positive and others negtive, then A is indefinite. Also note that if one of the charateristic roots is zero, the determinant will also be zero which implies that the matrix is

singular. Ina singular matrix, the rows and/or columns are linear combinations of each other. In other words, we are able to obtain a row or column of the matrix from another row or column by appropriate manipluations. Examine the following matrix:

> B:=matrix([[4,6],[6,9]]);

> detb:= det(B);

> with(linalg):
B1:=matrix([[4-lambda,6],[6,9-lambda]]);

> detb1:=det(B1);

> solve(detb1,lambda);

Since one of the roots is zero, and the other one is 13, the determinat will be zero. When the determinat is zero, the matrix does not have an inverse and hence we cannot obtain unique soultions for the x's. Note that the second column in matrix B is simply 3/4*first column.

>