Question: A data scientist models urban traffic flow with a transformation matrix $ \mathbf{M} $ that rotates traffic patterns by $ 45^\circ $ and scales them by $ \sqrt{2} $. Find $ \mathbf{M} $.

["Understanding the Transformation Matrix $ \mathbf{M} $ in Urban Traffic Flow Modeling", "When modeling urban traffic flow, data scientists often use linear transformations to represent complex movement patterns across city networks. A common operation involves rotating traffic flow data by a specific angle and scaling its magnitude—an essential step for simulating and predicting congestion patterns. In one such scenario, a transformation matrix $ \mathbf{M} $ rotates traffic flow patterns by $ 45^\circ $ and scales them uniformly by $ \sqrt{2} $. This article explains how to derive this matrix, blending geometric insight with practical application.", "### The Role of Rotation and Scaling in Traffic Modeling", "Traffic flows in a city can be represented as vectors in a 2D plane, where each vector captures both direction and intensity of vehicles moving between key intersections or zones. To analyze evolving traffic behavior, linear transformations—specifically rotation and scaling—allow data scientists to rotate directional patterns and adjust overall flow scale.", "Scaling by $ \sqrt{2} $ may reflect normalization or amplification of traffic intensity under certain conditions, while rotation by $ 45^\circ $ aligns traffic models with diagonal city layouts or directional preferences in urban infrastructure.", "### Deriving the Rotation and Scaling Matrix", "In linear algebra, a 2D transformation matrix combining rotation and uniform scaling takes the form:", "$$\n\mathbf{M} = \sqrt{2} \cdot \begin{bmatrix} \cos \ heta & -\sin \ heta \ \sin \ heta & \cos \ heta \end{bmatrix}\n$$", "Here, $ \ heta = 45^\circ $. We compute:", "$$\n\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\n$$", "Substituting:", "$$\n\mathbf{M} = \sqrt{2} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\n= \begin{bmatrix} \sqrt{2} \cdot \frac{\sqrt{2}}{2} & -\sqrt{2} \cdot \frac{\sqrt{2}}{2} \ \sqrt{2} \cdot \frac{\sqrt{2}}{2} & \sqrt{2} \cdot \frac{\sqrt{2}}{2} \end{bmatrix}\n$$", "Simplifying each entry:", "$$\n\sqrt{2} \cdot \frac{\sqrt{2}}{2} = \frac{2}{2} = 1\n$$", "Thus, the transformation matrix becomes:", "$$\n\mathbf{M} = \begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix}\n$$", "### Interpretation and Use in Traffic Flow Analysis", "This matrix $ \mathbf{M} $ rotates any input traffic flow vector counterclockwise by $ 45^\circ $ and scales its magnitude by $ \sqrt{2} \cdot |\mathbf{v}| $, effectively amplifying and realigning directional patterns. In practice, applying $ \mathbf{M} $ to traffic flow data helps correlate directional tendencies with spatial network flows—critical for optimizing traffic signals, modeling congestion propagation, and planning urban infrastructure.", "### Conclusion", "The matrix $ \mathbf{M} = \begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix} $ embodies the combined rotation and scaling transformation used by data scientists to model urban traffic flows. By encoding geometric rotation at $ 45^\circ $ with a $ \sqrt{2} $-scale factor, this matrix serves as a powerful tool for simulating real-world traffic dynamics in a mathematically precise and computationally efficient way.", "---", "Keywords: traffic flow model, data scientist, transformation matrix, rotation matrix 45 degrees, scaling factor $ \sqrt{2} $, linear transformation, urban traffic simulation"]