Create a diagonal matrix with entries "c." Multiply the scalar "c" by the identity matrix that has the same number of rows and columns as your covariance matrix., i.e., construct the matrix cI.
Subtract cI from the covariance matrix. The equation for this is C -- cI, assuming that "C" is your covariance matrix. The result will be a matrix that has pure numerical values only for off-diagonal elements.
Take the determinant of C -- cI. Use the method of calculating determinants on square matrices (C -- cI is a square matrix because its number of rows equal its number of columns). Call the determinant "D."
Set D equal to zero and solve the equation. Write D = 0. This is an equation with one variable, "c." Solve this equation using algebra, yielding multiple results for "c."
Observe the values of c. These values are the eigenvalues of the covariance matrix. Because you want the principal component (the eigenvector corresponding to the largest eigenvalue), you need to construct the eigenvector related to the largest c. Call the largest c "m."
Create the matrix C-mI. This is a square matrix similar to the covariance matrix but with different diagonal elements.
Multiply the matrix C-mI by a column vector of variables, "x." Create a column vector, "x," that has the same number of columns as C has. Perform the multiplication (C-mI)x. The result will be a column vector of polynomials.
Set (C-mI)x equal to zero and solve for x using matrix algebra. The solution for x is the eigenvector of interest: the principal component for the covariance matrix.