Is the main diagonal value of the upper triangular matrix its eigenvalue? What about the lower triangular matrix?

Is the main diagonal value of the upper triangular matrix its eigenvalue? What about the lower triangular matrix?

Let n-order upper triangular square matrix A, whose eigenvalue is λ. According to the calculation formula of the eigenvalue of the matrix, there is | a - λ e | = 0, then there is: | a11 - λ A12 A13 a1n| | a22-λ a23 a24 ……… a2n|| a33-λ ………………… a3n|=0|…………………………………… || an-...

The sum of eigenvalues of a matrix is equal to the sum of main diagonal elements, and the product of eigenvalues is equal to the product of main diagonal elements. Why?
For a particular matrix or for all matrices?

It seems that you asked both sides
These two sentences depend on the fact that a matrix has n eigenvalues
Similarity, trace and determinant are the same, which does not depend on the n eigenvalues of the matrix and their diagonalization

Using C language to find the sum of diagonal elements of an integer matrix with three rows and three columns
C language

#include
main()
{
int i,j,sum=0,n=0;
int a[3][3];
//Create matrix
Printf ("please enter the matrix to be calculated): (n));
for(i=0;i

This is a C programming problem, find a 3 * 3 matrix diagonal elements

main()
{
int x[3][3]={0};
Int a = 0, B = 0; / / are two diagonal lines and
int i,j;
for(i=0;i

Find the sum of diagonal elements of a 3 * 3 matrix? (how to code with C language)
Double for loop control is used to input two-dimensional array a, and then a [i] [i] is accumulated and output

#include
void main()
{
int a[3][3];
int i,j;
int sum=0;
int sum1=0;
// input
for( i=0; i

Using C language to find the sum of diagonal elements of a 3 * 3 matrix

#include
int main(){
float st[3][3],sum=0;
int i,j;
for(i=0;ist[i][j];
for(i=0;i

Find a 5 * 5 integer matrix diagonal elements, using C language

for(i=0;i=0;i--,j++)
sum1=sum1+a[i][j];

If there is an orthogonal matrix P, what's the difference between using P ^ - 1 * a * P or P ^ t * a * P to find the diagonalization matrix Λ of a, and using the second method to find the diagonalization matrix Λ which is not the eigenvalue of a
Using the method of | a - λ e | to find the eigenvalues of a, the diagonal matrices Λ, P1, P2 When p ^ - 1 * a * P = ∧, but sometimes p ^ t * a * P ≠∧

1. C is invertible and C ^ TAC is a diagonal matrix, which is a congruent transformation
For example, using the matching method
2. P = (α 1, α 2,... α n), α 1, α 2,... α n are the linearly independent eigenvectors of a belonging to the eigenvalues λ 1, λ 2,..., λ n respectively
Here p ^ - 1AP = diag (λ 1, λ 2,..., λ n), which is similar diagonalization
3. Q = (α 1, α 2,... α n), α 1, α 2,... α n are two orthogonal eigenvectors of a with length 1 belonging to eigenvalues λ 1, λ 2,..., λ n respectively
In this case, Q ^ - 1aq = diag (λ 1, λ 2,..., λ n), which is orthogonal similar diagonalization
Only in this case, Q ^ - 1aq = q ^ Taq
Note: when a is a symmetric matrix, the eigenvectors of multiple eigenvalues should be orthogonalized and all eigenvectors should be unitized

For example, a is a matrix. After calculation, the eigenvalues of a are 1,3, - 1, but how can the diagonalized diagonal matrix be sorted from the upper left corner to the lower right corner

Let's see how you diagonalize the invertible matrix. According to the eigenvectors (column vectors) corresponding to 1,3, - 1 from left to right, we get the invertible matrix P of order 3. Diagonalizing a with this p is diag {1,3, - 1}. Similarly, if you are diag {1,3, - 1} according to the eigenvectors (column vectors) corresponding to 3,1, - 1

It is known that when a matrix is diagonalized, the eigenvalues on the main diagonal are eigenvalues. What is the order of these eigenvalues, or does the order of the eigenvalues affect the result of diagonalization?

There is no order requirement. Just note that in P ^ (- 1) AP = B (a is the matrix to be diagonalized, B is the diagonal matrix), the position of the column vector of P (i.e. the eigenvector of a) in P ^ (- 1) AP = B should correspond to the position of the eigenvalue in B one by one
The similarity canonical form of a is uniquely determined except for the order of elements on the main diagonal