The characteristic polynomial of a square matrix is the determinant of the difference of the matrix and a variable times the identity matrix.  If A is a square matrix and v is a variable, CHARPOLY(A, v) simplifies to the characteristic polynomial of A in terms of v.  For example,

CHARPOLY([2, 3; a, b], z)

simplifies to

 2                         
z  - z·(b + 2) - 3·a + 2·b 


The default value of the variable used by CHARPOLY is w.


The eigenvalues of a square matrix are the zeros of its characteristic polynomial.  If A is a square matrix and v is a variable, EIGENVALUES(A, v) simplifies to a vector of the eigenvalues of A in terms of v.  For example,

EIGENVALUES([2, 3; 0, b], z)

simplifies to [z = 2, z = b].  If no variable is given, a vector of just the eigenvalues is returned.


The degree of the characteristic polynomial equation that EIGENVALUES must solve is equal to the dimension of the matrix.  Since characteristic polynomials rarely factor exactly over the rational numbers, exact results usually require the cubic formula for 3 by 3 matrices or the quartic formula for 4 by 4 matrices.  Exact results are usually impossible for matrices larger than 4 by 4.  Moreover, the quartic formula generates such bulky results that the exact eigenvalues of 4 by 4 matrices often exhaust memory.  Thus, exact eigenvalues are rarely attainable for matrices larger than 3 by 3.


The characteristic polynomial of a numerical matrix is univariate.  Thus real and complex eigenvalues can be approximated by finding the zeros of the characteristic polynomial using the Solve > Expression command1PET_XA.  Note that the eigenvalues of symmetric and Hermitian matrices are all real.


The file LinearAlgebra.mthLinear_Algebra defines functions for computing exact and approximate eigenvectors.


Other Vectors and MatricesVectors_and_Matrices topics

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