> For the complete documentation index, see [llms.txt](https://ykkim.gitbook.io/ec/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ykkim.gitbook.io/ec/numerical-programming/ta-tutorial/tutorial-eigenvalue-problem.md).

# Tutorial: Eigenvalue problem

## Tutorial: MATLAB

Estimate the eigenvalue and eigenvectors

```matlab
clear; close all;

A = [45 30 -25; 30 -24 68; -25 68 80];

disp('Eigvalue and vector of A (MATLAB):');
[eigVec,eigVal]=eig(A)
eigvalues=diag(eigVal)
```

## Exercise 1: MATLAB

**Download the tutorial source file**

* [TU\_Eigenvalue\_Student\_2025.mlx](https://github.com/ykkimhgu/NumericalProg-student/blob/main/tutorial/TU_Eigenvalue/TU_Eigenvalue_Student_2025.mlx)

**Fill-In the blanks.**

```matlab
function [Q, R] = QRdecomp_student(A)  
% Factorization of given matrix [A] into Upper triagular [R] and orthogonormal [Q]
% Using HouseHold Matrix
% Input: A (nxn)
% Output: Q (nxn), R (nxn)
    
    % Initialization
    n = size(A,1);
    I=eye(n);
    H=zeros(n,n);
    R=A;
    Q = I;    
    
    for j = 1:n-1                
        % Step 1. Create vector [c]
        % [YOUR CODE GOES HERE]
        % c = _______________;   
        

        % Step 2. Create vector [e]
        e=zeros(n,1);
        % [YOUR CODE GOES HERE]
        % e = _______________;
        

        % Step 3. Create vector [v]
        % [YOUR CODE GOES HERE], HINT: use norm(c,2)
        % v = _______________;
    

        % Step 4. Create matrix [H]
        % [YOUR CODE GOES HERE]
        % H = _______________;
        
        % Step 5. Update [Q], [R]
        Q = Q*H;
        R = H*R;
    end

end % end of function
```

**Run the code and check the answer with MATLAB's eig(A)**

```matlab
% initialize  
N=100;
U=A; 

for i = 1:N 
    % Step 1: A=QR decomposition
    [Q, R] = QRdecomp_student(U); 
    
    % Step 2: Create Similar Matrix A = Q'AQ
    U = R*Q;   
    if(~mod(i,10))                
        disp(sprintf('iteration %d \n',i));
        U
    end
end

% Step 3: eigenvalues from U
lamdas = diag(U);
```

## Exercise 2: Eigenvalue in C-Programming

**Download the tutorial source file**

* [Tutorial\_Eigenvalue\_Student.cpp](https://github.com/ykkimhgu/NumericalProg-student/blob/main/tutorial/TU_Eigenvalue/TU_Eigenvalue_Student.cpp)

**Create the function that returns the estimated eigenvalues**

```c
Matrix eigval_student(Matrix _A); // returns Nx1 vector
void QRdecomp_student(Matrix Q, Matrix R, Matrix _A);


// Usage example
Matrix eigVals = eigval_student(matA);  
```

## Exercise 3: Eigenvector in C-Programming

```c
Matrix eigvec_student(Matrix _A);
// Usage example
Matrix eigVec = eigvec_student(matA);
```