Question

Find the QR factorization of the general \(2 \times 2\) matrix $$ A=\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) $$

Short answer

Question: Find the QR factorization of a general \(2 \times 2\) matrix A using the Gram-Schmidt process. Answer: The QR factorization of matrix \(A\) can be represented as: $$ A = \left(\begin{array}{ll} \frac{a_{11}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \\ \frac{a_{21}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \end{array}\right) \left(\begin{array}{ll} \sqrt{a_{11}^2 + a_{21}^2} & \frac{a_{11}a_{12}+a_{21}a_{22}}{\sqrt{a_{11}^2 + a_{21}^2}} \\ 0 & \sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2} \end{array}\right) $$ The matrix \(A\) is factored into an orthogonal matrix Q and an upper triangular matrix R.

Step-by-step solution

Define the given matrix A

Write down the given matrix \(A\). $$ A=\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) $$

Apply the Gram-Schmidt process for the first column

Let's apply the Gram-Schmidt process to the first column vector in A. Let \(u_1\) denote the first column and \(v_1 = u_1\). $$ v_1 = u_1 = \left(\begin{array}{l} a_{11} \\ a_{21} \end{array}\right) $$

Normalize the first column vector

Next, we need to normalize the first column vector \(v_1\). Let's denote the normalized vector as \(e_1\). $$ e_1 = \frac{v_1}{\|v_1\|} = \frac{1}{\sqrt{a_{11}^2 + a_{21}^2}} \left(\begin{array}{l} a_{11} \\ a_{21} \end{array}\right) $$

Apply the Gram-Schmidt process for the second column

Now, let's apply the Gram-Schmidt process to the second column vector in A. Let \(u_2\) denote the second column. $$ u_2 = \left(\begin{array}{l} a_{12} \\ a_{22} \end{array}\right) $$ Project \(u_2\) onto \(e_1\), which results in the following vector: $$ proj_{e_1}(u_2) = (u_2 \cdot e_1)e_1 = \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}\left(\begin{array}{l} a_{11} \\ a_{21} \end{array}\right) $$ Obtain the orthogonal vector \(v_2\) by subtracting the projection from \(u_2\). $$ v_2 = u_2 - proj_{e_1}(u_2) = \left(\begin{array}{l} a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11} \\ a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21} \end{array}\right) $$

Normalize the second column vector

Normalize the second column vector \(v_2\). Let's denote the normalized vector as \(e_2\). $$ e_2 = \frac{v_2}{\|v_2\|} = \frac{1}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \left(\begin{array}{l} a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11} \\ a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21} \end{array}\right) $$

Form the Q matrix

Now form the matrix \(Q\) using normalized column vectors \(e_1\) and \(e_2\). $$ Q = \left(\begin{array}{ll} e_1 & e_2 \end{array}\right) = \left(\begin{array}{ll} \frac{a_{11}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \\ \frac{a_{21}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \end{array}\right) $$

Compute the entries for the matrix R

Finally, compute the entries for the upper triangular matrix \(R\). This can be done using the dot product of the normalized vectors and the original column vectors. $$ R = \left(\begin{array}{ll} r_{11} & r_{12} \\ 0 & r_{22} \end{array}\right) = \left(\begin{array}{ll} e_1 \cdot u_1 & e_1 \cdot u_2 \\ 0 & e_2 \cdot u_2 \end{array}\right) = \left(\begin{array}{ll} \sqrt{a_{11}^2 + a_{21}^2} & \frac{a_{11}a_{12}+a_{21}a_{22}}{\sqrt{a_{11}^2 + a_{21}^2}} \\ 0 & \sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2} \end{array}\right) $$

Express A as QR

Now, we have found Q and R matrices, so we can express A as the product of these matrices: $$ A = QR = \left(\begin{array}{ll} \frac{a_{11}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \\ \frac{a_{21}}{\sqrt{a_{11}^2 + a_{21}^2}} & \frac{a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21}}{\sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2}} \end{array}\right) \left(\begin{array}{ll} \sqrt{a_{11}^2 + a_{21}^2} & \frac{a_{11}a_{12}+a_{21}a_{22}}{\sqrt{a_{11}^2 + a_{21}^2}} \\ 0 & \sqrt{(a_{12} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{11})^2 + (a_{22} - \frac{a_{11}a_{12}+a_{21}a_{22}}{a_{11}^2 + a_{21}^2}a_{21})^2} \end{array}\right) $$

Key concepts

Gram-Schmidt Process

The Gram-Schmidt process is a method used in linear algebra to orthogonalize a set of vectors in an inner product space. It transforms a set of linearly independent vectors into a set of orthogonal vectors. Here’s how it works. Start with a set of vectors, for instance, columns of a matrix. For each vector, you adjust it by removing the components aligned with previously calculated vectors.
This process is like pulling out any overlapping parts with other vectors, one by one, ensuring all vectors eventually stand in unique directions. 🎨
In our QR factorization example, we used the Gram-Schmidt process to make the columns of matrix A orthogonal. We took the first column and used it as our base vector. For the second column, we projected it onto the first to find its overlapping part. We then subtracted this overlap to get a new vector, resulting in two orthogonal vectors.
By this orthogonalization, these vectors can later be normalized to form the Q matrix part of the QR factorization.

Orthogonalization

Orthogonalization is about making vectors at right angles to each other, similar to setting train tracks so they never converge.
In linear algebra, two vectors are orthogonal if their dot product is zero. Orthogonal vectors are very useful because they make calculations easier and more stable, especially in numerical settings.
During QR factorization, orthogonalization can be visualized in the Gram-Schmidt process. After de-overlapping vectors, you ensure your vectors are orthogonal before normalizing them.
This results in the columns of matrix Q being orthogonal unit vectors. They span the same subspace as the original matrix columns but have the desirable property of orthogonality. In simple terms, they form a perfect grid!

• Orthogonal vectors simplify complex calculations.
• They are foundational in many areas, including signal processing and computer graphics.

Matrix Decomposition

Matrix decomposition refers to breaking down a matrix into simpler, more manageable pieces, often to simplify complex matrix operations. QR factorization is a type of matrix decomposition that expresses a matrix as the product of an orthogonal matrix (Q) and an upper triangular matrix (R).
A key benefit of matrix decomposition lies in its wide variety of applications, including solving linear equations, performing linear regression, and eigenvalue calculations.
In assessments or computational tasks, breaking down matrices in this manner can transform a seemingly complex challenge into several simpler tasks. This makes them much easier to tackle with the help of smaller, more manageable computations.
Matrix decompositions like QR are crucial in numerical linear algebra as they provide robust techniques to solve numerical problems efficiently.

Numerical Linear Algebra

Numerical linear algebra studies algorithms to solve problems in linear algebra on computers. Unlike pure linear algebra, numerical linear algebra must deal with real-world complications, like inaccuracies in floating-point arithmetic.
It often employs matrix decompositions, such as QR factorization, to enhance stability and efficiency in computations.
For example, systems of linear equations, eigenvalue problems, and matrix factorizations are better solved numerically using stable algorithms that counteract computational errors.
With these advanced techniques, computers can handle large data sets often encountered in fields such as data science, engineering, and machine learning. So, numerical linear algebra is like turning tricky math into a practical toolkit for modern technology.