A is a matrix of order n. It is proved that tr (a ^ k) = the sum of the k-th power of the eigenvalues of A
Let A1,..., an be the eigenvalue of A
Then A1 ^ k,..., an ^ k is the eigenvalue of a ^ k (theorem conclusion)
So tr (a ^ k) = A1 ^ k +... + an ^ K. (theorem)
RELATED INFORMATIONS
- 1. Let a matrix of order 4 satisfy | 3e-a |, AAT = 2E, | a|
- 2. The eigenvalues of adjoint matrix A * are 1,2,4,8. Find the eigenvalues of (1 / 3a) ^ - 1
- 3. Let a be a square matrix of order n, 2,4,..., 2n be the n eigenvalues of a, and calculate the value of determinant / a-3e /
- 4. Let a be a square matrix of third order, and it is known that a has two eigenvalues - 1. - 2, and the rank of (a + 3e) is 2. Find the determinant of a + 4E
- 5. Let the three eigenvalues of a matrix of order 3 be 2, - 4,3, then the three eigenvalues of a * are () A 2,-4,3 B 1/2,-1/4,1/3 C -12,6,-8 D 12,-6,8
- 6. Let 2 be an eigenvalue of a square matrix of order 3, then a ^ 2 must have an eigenvalue?
- 7. Let the three eigenvalues of a matrix of order 3 be - 1,2,4, then what are the three eigenvalues of a *?
- 8. 4. Let the four eigenvalues of a square matrix of order 4 be 3,1,1,2, then | a|=
- 9. Let a be a square matrix of order n and satisfy a ^ 2-3a + 2E = 0, it is proved that the eigenvalue of a can only be 1 or 2
- 10. If a is a square matrix of order n and AAT = e, | a | = - 1, it is proved that | a + I | = 0, where I is the identity matrix
- 11. If the eigenvalue of matrix A is t, why is the eigenvalue of K power of a K power of T,
- 12. Let a matrix of order n have n eigenvalues 0,1,2,..., n-1, and B ~ A, and find det (I + b)
- 13. Given that the matrix A of order n satisfies a ^ 2-2a-3e = 0, it is proved that the eigenvalue of a can only be - 1 or 3. How to prove that it can only be - 1? (- e-A) (3e-a) = 0, but how can we prove that it can only be - 1 or 3?
- 14. Let a be a matrix of order n, | a | ≠ 0, a * be the adjoint matrix of a, and E be the unit matrix of order n. if a has an eigenvalue λ, then (a *) 2 + e must have an eigenvalue______ .
- 15. If the 3-dimensional column vectors α and β satisfy α 'β = 2, how to find the nonzero eigenvalues of the matrix β α'?
- 16. Let a be a square matrix of order 2, a, β be linearly independent 2-dimensional sequence vectors, AA = 0, a β = a + β, then the nonzero eigenvalues of A
- 17. If the 3-dimensional column vectors α and β satisfy α t β = 2, then the nonzero eigenvalue of the matrix β α t is? Although the characteristic polynomials and eigenvalues of AB and Ba are the same, why is the nonzero eigenvalue equal to 2? To Yin Yang double edged sword: β α t β = β (α t β) = 2 β or not
- 18. A third-order matrix has only two linearly independent eigenvectors, and this matrix has only one eigenvalue of the triple root
- 19. Let a be a matrix of order 2, α 1 and α 2 be two linearly independent two-dimensional vectors, a α 1 = O, a α 2 = 2 α 1 + α 2, and find the non-zero eigenvalues of A
- 20. Let a be a matrix of order n and satisfy AAT = e. the determinant of a is less than zero. It is proved that - 1 is an eigenvalue of A