Higher algebra A is a matrix of order n over a complex field, R1, R2,..., RN are all eigenvalues of a (multiple roots are calculated by multiplicity) proof (1) if f (x) f (R1) The higher algebra A is a matrix of order n over a complex field. R1, R2,..., RN are all the eigenvalues of A. This paper proves that (1) if f (x) is a polynomial of degree greater than 0 on C, then f (R1), f (R2),... F (RN) are all the eigenvalues of F (a). (2) if a is invertible, 1 / R1, 1 / R2,..., 1 / RN are all the eigenvalues of a ^ - 1

Higher algebra A is a matrix of order n over a complex field, R1, R2,..., RN are all eigenvalues of a (multiple roots are calculated by multiplicity) proof (1) if f (x) f (R1) The higher algebra A is a matrix of order n over a complex field. R1, R2,..., RN are all the eigenvalues of A. This paper proves that (1) if f (x) is a polynomial of degree greater than 0 on C, then f (R1), f (R2),... F (RN) are all the eigenvalues of F (a). (2) if a is invertible, 1 / R1, 1 / R2,..., 1 / RN are all the eigenvalues of a ^ - 1

Because a is a matrix of order n over a complex field, R1, R2... And RN are all the eigenvalues of a (multiple roots are calculated by multiplicity),
So the main diagonal elements of Jordan canonical form of a are R1, R2... RN
(1) If f (x) is a polynomial of degree greater than 0 on C, then the main diagonal elements of Jordan canonical form of F (a) are
F(R1),F(R2),...F(RN)
It can be seen that f (R1), f (R2),... F (RN) are all characteristic roots of F (a)
(2) If a is reversible, then R1, R2... And RN are nonzero, and the main diagonal elements of Jordan canonical form of a ^ - 1 are zero
1/R1,1/R2,...,1/RN,
It can be seen that 1 / R1, 1 / R2,..., 1 / RN are all the eigenvalues of a ^ - 1