Introduction to Matrix QR Decomposition Calculator:
QR decomposition calculator is a great source that helps you to find the QR decomposition of a given matrix. Our tool determines the orthogonal matrix Q and the upper triangular matrix R that combine to form decomposition.
It is a valuable tool because QR decomposition is a complicated method in vector space. When you do manual calculations for learning purposes and practical work you must try our q r decomposition calculator for quick and easy solutions.
What is the QR Decomposition Method?
QR decomposition is a process that is used in linear algebra to decompose a matrix into the product of an orthogonal matrix Q and an upper triangular matrix R such as A = QR. It is known as QR decomposition or QR factorization.
QR decomposition is a fundamental method in numerical linear algebra, providing a structured way to analyze and solve various matrix problems efficiently.
This decomposition is widely used in numerical computations, for solving systems of linear equations, least squares, eigenvalues, and various other applications.
How to Find the QR Decomposition of a Matrix?
To find the QR decomposition of a matrix we use the Gram-Schmidt orthogonalization process to get Q which is the orthogonality and rank of matrix R. Here are the steps that show how to find the QR Decomposition problem in step for better understanding.
QR Decomposition Example: Find the QR Decomposition of a Matrix
$$ \begin{matrix} 1 & -1 & 4 \\ 1 & 4 & -2 \\ 1 & 4 & 2 \\ \end{matrix} $$
Solution:
Start with the given matrix A.
$$ Here\; A \;=\; \left[ \begin{matrix} 1 & -1 & 4 \\ 1 & 4 & -2 \\ 1 & 4 & 2 \\ \end{matrix} \right] $$
Let's take q1=a1,
$$ q_1’ \;=\; a_1 \;=\; \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] $$
Find the magnitude of r11,
$$ r_{11} \;=\; ||q_1’ || \;=\; \sqrt{1^2 + 1^2 + 1^2 } \;=\; \sqrt{3} \;=\; 1.7321 $$
For q1 value,
$$ q_1 \;=\; \frac{1}{||q_1’||} . q_1’ \;=\; \frac{1}{1.7321} . \left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0.5774 \\ 0.5774 \\ 0.5774 \\ \end{matrix} \right] $$
For r12: take the dot product of q1 with a2,
$$ r_{12} \;=\; q_1^T . a_2 \;=\; [\begin{matrix} 0.5774 & 0.5774 & 0.5774 \\ \end{matrix}] \times \left[ \begin{matrix} -1 \\ 4 \\ 4 \\ \end{matrix} \right] \;=\; 4.0415 $$
To find q2 we find the magnitude of q2.
$$ q_2' \;=\; a_2 - r_{12} . q_1 \;=\; \left[ \begin{matrix} -1 \\ 4 \\ 4 \\ \end{matrix} \right] -4.0415 \left[ \begin{matrix} 0.5774 \\ 0.5774 \\ 0.5774 \\ \end{matrix} \right] \;=\; \left| \begin{matrix} -\frac{10}{3} \\ \frac{5}{3} \\ \frac{5}{3} \\ \end{matrix} \right| $$
$$ r_{22} \;=\; ||q_2’|| \;=\; \sqrt{\left( -\frac{10}{3} \right)^2 + \left(\frac{5}{3} \right)^2 + \left(\frac{5}{3} \right)^2} \;=\; \sqrt{\frac{50}{3}} \;=\; 4.0825 $$
$$ q_2 \;=\; \frac{1}{||q_2’} . q_2’ \;=\; \frac{1}{4.0825} . \left[ \begin{matrix} -\frac{10}{3} \\ \frac{5}{3} \\ \frac{5}{3} \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} -0.8165 \\ 0.4082 \\ 0.4082 \\ \end{matrix} \right] $$
Compute r13,
$$ r_{13} \;=\; q_1^T . a_3 \;=\; [\begin{matrix} 0.5774 & 0.5774 & 0.5774 \\ \end{matrix}] \times \left[ \begin{matrix} 4 \\ -2 \\ 2 \\ \end{matrix} \right] \;=\; 2.3094 $$
Compute r23 to find q`3,
$$ r_{23} \;=\; q_2^T . a_3 \;=\; [ \begin{matrix} -0.8165 & 0.4082 & 0.4082 \\ \end{matrix} ] \times \left[ \begin{matrix} 4 \\ -2 \\ 2 \\ \end{matrix} \right] \;=\; -3.266 $$
$$ q_3’ \;=\; a_3 - r_{13} . q_1 - r_{23} . q_2 \;=\; \left[ \begin{matrix} 4 \\ -2 \\ 2 \\ \end{matrix} \right] -2.3094 \left[ \begin{matrix} 0.5774 \\ 0.5774 \\ 0.5774 \\ \end{matrix} \right] + 3.266 \left[ \begin{matrix} -0.8165 \\ 0.4082 \\ 0.4082 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ -2 \\ 2 \\ \end{matrix} \right] $$
Find the magnitude of r33 to get q3,
$$ r_{33} \;=\; ||q_3’|| \;=\; \sqrt{(0)^2 + (-2)^2 + (2)^2} \;=\; \sqrt{8} \;=\; 2.8284 $$
$$ q_3 \;=\; \frac{1}{||q_3’||} . q_3’ \;=\; \frac{1}{2.8284} . \left[ \begin{matrix} 0 \\ -2 \\ 2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ -0.7071 \\ 0.7071 \\ \end{matrix} \right] $$
Now use all the above value of q1,q2,q3 and form the matrix Q,
$$ Q \;=\; [q_1, q_2, q_3] \;=\; \left[ \begin{matrix} 0.5774 & -0.8165 & 0 \\ 0.5774 & 0.4082 & -0.7071 \\ 0.5774 & 0.4082 & 0.7071 \\ \end{matrix} \right] $$
Use the upper triangle method to get the value of R,
$$ R \;=\; \left[ \begin{matrix} r_{11} & r_{12} & r_{13} \\ 0 & r_{22} & r_{23} \\ 0 & 0 & r_{33} \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 1.7321 & 4.0415 & 2.3094 \\ 0 & 4.0825 & -3.266 \\ 0 & 0 & 2.8284 \\ \end{matrix} \right] $$
To verify the QR decomposition, commute A = Q x R,
$$ Q \times R \;=\; \left[ \begin{matrix} 0.5774 & -0.8165 & 0 \\ 0.5774 & 0.4082 & -0.7071 \\ 0.5774 & 0.4082 & 0.7071 \\ \end{matrix} \right] \times \left[ \begin{matrix} 1.7321 & 4.0415 & 2.3094 \\ 0 & 4.0825 & -3.266 \\ 0 & 0 & 2.8284 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 1 & -1 & 4 \\ 1 & 4 & -2 \\ 1 & 4 & 2 \\ \end{matrix} \right] $$
Hence we get the matrix A again that means our solution is correct.
How to Use the QR Calculator?
QR decomposition online calculator has a simple design that enables you to use it to easily calculate qr decomposition of the matrix questions. Before adding the input value of the matrix for solutions, you must follow our instructions. These instructions are:
- Add the size of the matrix in the given box.
- Enter the elements of the matrix in the input box.
- Recheck your input matrix value before hitting the calculate button to start the calculation process in the qr decomposition calculator with steps.
- Click on the “Calculate” button to get the desired result of your given QR Decomposition question.
- If you want to try out our q r decomposition calculator for the first time then you can use the load example.
- Click on the “Recalculate” button to get a new page for solving more QR Decomposition problems to get solutions.
Final Result of QR Decomposition Online Calculator:
QR calculator gives you the solution to a given matrix problem when you add the input to it. It provides you with solutions to the QR decomposition problem. It may contain as:
- Result option
You can click on the result option and it provides you with a solution for the QR Decomposition questions.
- Possible step
When you click on the possible steps option it provides you with the solution of QR decomposition problems in steps.
Useful Features of QR Decomposition Calculator:
Q R decomposition calculator has many useful features that you obtain whenever you use it to calculate decomposition problems for finding the orthogonality and upper triangle matrix in solutions. These features are:
- It is an adaptable tool that can operate through electronic devices like laptops, computers, mobile, tablets, etc with the help of the internet.
- It is a free tool so you can use it to find the QR decompostion problems in the qr decomposition online calculator.
- Our tool saves the time and effort that you consume in doing lengthy calculations of QR decomposition in a few seconds.
- It is a learning tool so you can use this qr calculator for practice so that you get in-depth knowledge of this method.
- It is a trustworthy tool that provides you with accurate solutions according to your input matrix value whenever you use it to get a solution.
- QR decomposition calculator provides you solutions of QR decomposition questions with a complete process in a step-by-step method for a better understanding.
