How to prove the existence of orthogonal matrix t to make t'at a triangular matrix

This conclusion is not enough
1. Eigenvectors belonging to different eigenvalues are orthogonal
2. For the k-fold eigenvalue a of a, there are k linearly independent eigenvectors
(the key point of this conclusion is to ensure that a can be diagonalized, and then from 1 to 1.)
The first proof is simpler, and the second is troublesome. Textbooks generally don't prove it

RELATED INFORMATIONS

1. How to prove that the addition and scalar multiplication of all upper triangular matrices are linear spaces in real number field
2. Two polynomials are not divisible in rational number field, but divisible in complex number field? It is helpful for the responder to give an accurate answer
3. x^n+x^(n-1) ………… Factorization of X + 1 in complex number field and real number field
4. The fifth power of polynomial x-9xy is decomposed into rational number range, real number range and complex number range in the following range
5. A is a matrix of order n, α 1, α 2 α n is n-dimensional sequence vector, α n ≠ 0, a α 1 = α 2 ,Aαn-1=αn,Aα
6. The even power and the odd power of a number are opposite to each other______ A. 1 B. - 1 C.1 or 0 D. - 1 or 0
7. If the even power of a rational number is positive, then the rational number must be positive. Right or wrong?
8. The even power of all rational numbers is ()
9. The base of (- 2) 3 is______ The index is______ The power is______ ．
10. -The base of 3 ^ 2 is () the exponent is () and the power is () The key is what is the power?
11. How to solve polynomial equation on complex field by numerical method I have a question to ask you, that is, how to write a program to solve the polynomial equation in the complex field (for example, f (x) = A2 * x ^ 2 + A1 * x + A0 = 0). Now I use the argument principle (in short, I integrate f '(x) / F (x) on the complex plane. If there is an internal solution, the integral is not zero, otherwise it is zero), The solution is determined by continuously narrowing the integral range. However, for the higher order (greater than 10 orders), it is not enough. Is there any algorithm improvement that can improve the accuracy and speed?
12. X ^ 4 + 1 is decomposed into irreducible polynomials over the rational field
13. It is proved that irreducible polynomials with real roots in rational number field must be quadratic polynomials
14. It is proved that there exists an irreducible polynomial f (x) over the field of rational numbers, such that f (a) = 0
15. Find I + root 2 in the rational number field Q irreducible polynomial, you master please tell me
16. 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
17. Let a be a matrix of order n over the complex field C It is proved that there exists an invertible matrix P of order n on C such that P ^ - 1AP = R1 A12. A1N 0 a22 .a2n . 0 an2 .ann
18. On the proof of matrix complex field, we will add 1-2 times points Let a be a matrix of order n over a complex field 1) A is similar to a matrix in the form of: λ1 c12 c13 ... c1n 0 λ2 c23 ... c2n 0 0 λ3 ... c3n ... ... ... ... ... 0 0 0 ... λn 2) A has n eigenvalues (multiple roots count multiple numbers) in complex field, and if λ 1, λ 2,..., λ n are all eigenvalues and f (x) is any polynomial in complex field, then f (λ 1), f (λ 2),..., f (λ n) are all eigenvalues of F (a)
19. Use Maple to solve the equation x^5+x^4−12x^3−21x^2+x+5=0 y^5+y^4−16y^3+5y^2+21y−9=0 y^5+y^4−24y^3−17y^2+41y−13=0 y^5+y^4−28y^3+37y^2+25y+1=0 10x^6-75x^3-190x+21=0 All the previous quintic equations have radical solutions The last one, I don't know if there is. If there is, we need radical solution No, I hope it can be solved with a special function, like this z^5+z^4-e^6=0 z=exp(6/5)*hypergeom([-1/5,1/20,3/10,11/20],[1/5,2/5,3/5],256/3125*exp(-6)) +2/25*exp(-6/5)* hypergeom([1/5,9/20,7/10,19/20],[3/5,4/5,7/5],2/3125*exp(-6)) -4/125*exp(-12/5)* hypergeom([2/5,13/20,9/10,23/20],[4/5,6/5,8/5],256/3125*exp(-6)) +7/625*exp(-18/5)* hypergeom([3/5,17/20,11/10,27/20],[6/5,7/5,9/5],256/3125*exp(-6)) -1/5
20. Tan α = 2, find the value of [(sin α + cos α) / (sin α + cos α)] + cos & # 178; α