Solution: A rotation matrix for $ \theta = 45^\circ $ is $ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $. Scaling by $ \sqrt{2} $ multiplies each entry:

["Optimizing 2D Transformations: The Scaled Rotation Matrix for 45°", "Understanding how to properly apply rotations in 2D space is fundamental in graphics, robotics, physics, and engineering. One elegant and efficient way to represent a rotation in the plane is through a rotation matrix. This article explores the standard rotation matrix at a 45-degree angle and explains the important mathematical trick of scaling it by ( \sqrt{2} ), which enhances numerical precision and simplifies many geometric computations.", "---", "### The Standard Rotation Matrix at ( \ heta = 45^\circ )", "A rotation matrix for an angle ( \ heta ) around the origin in 2D is given by:", "[\nR(\ heta) = \begin{bmatrix} \cos\ heta & -\sin\ heta \ \sin\ heta & \cos\ heta \end{bmatrix}\n]", "For ( \ heta = 45^\circ ), we use the exact trigonometric values:", "[\n\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\n]", "Thus, the rotation matrix becomes:", "[\nR(45^\circ) = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\n]", "This matrix accurately rotates any 2D vector counterclockwise by 45 degrees about the origin.", "---", "### Scaling the Rotation Matrix by ( \sqrt{2} )", "An insightful mathematical transformation involves scaling the rotation matrix by ( \sqrt{2} ). When each entry is multiplied by ( \sqrt{2} ), the resulting matrix is:", "[\n\sqrt{2} \cdot R(45^\circ) = \sqrt{2} \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\n]", "Multiplying each term:", "[\n= \begin{bmatrix} \sqrt{2} \cdot \frac{\sqrt{2}}{2} & \sqrt{2} \cdot \left(-\frac{\sqrt{2}}{2}\right) \ \sqrt{2} \cdot \frac{\sqrt{2}}{2} & \sqrt{2} \cdot \frac{\sqrt{2}}{2} \end{bmatrix}\n= \begin{bmatrix} \frac{2}{2} & -\frac{2}{2} \ \frac{2}{2} & \frac{2}{2} \end{bmatrix}\n= \begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix}\n]", "Thus, scaling the original ( 45^\circ ) rotation matrix by ( \sqrt{2} ) yields:", "[\n\begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix}\n]", "---", "### Why This Scaling Matters", "This scaled matrix is not just a mathematical curiosity—it provides practical advantages:", "- Integer Entries: Unlike irrational values involving ( \frac{\sqrt{2}}{2} ), this matrix uses whole numbers, making manual computation and programming simpler and faster.", "- Preserved Rotation Property: Even scaled, the matrix remains an orthogonal transformation with determinant ( 2 ), but scaled such that rotation behavior is enhanced for specific applications like floating-point alignment.", "- Numerical Stability: Integer entries minimize rounding errors in computer arithmetic, improving stability in iterative calculations.", "---", "### Geometric Interpretation", "Geometrically, the original rotation matrix rotates vectors using scaling factor ( \frac{\sqrt{2}}{2} \approx 0.707 ). Scaling by ( \sqrt{2} ) amplifies each coordinate-wise operation, effectively transforming the rotation into a "simple" integer-coefficient matrix without sacrificing rotational accuracy.", "This trick is particularly useful in:", "- Computer Graphics: Simplifying rotation operations in pixel-level rendering\n- Game Development: Fast and stable coordinate transformations\n- Robotics: Matrix-based kinematic transformations", "---", "### Conclusion", "The 45-degree rotation matrix", "[\n\begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\n]", "becomes elegantly simple when scaled by ( \sqrt{2} ), yielding the integer matrix", "[\n\begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix}\n]", "This scaled form offers computational efficiency and numerical robustness, making it a powerful tool in dynamic 2D transformations. Leverage this insight to simplify matrix operations and improve performance across imaging, simulation, and control systems.", "---", "Keywords: rotation matrix, 2D rotation, 45 degree matrix, scaling by √2, orthogonal transformation, computer graphics matrices, numerical stability in matrices, linear algebra transformations."]