Matrix–Vector Multiplication Calculator

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Matrix–vector multiplication forms a new vector \( \mathbf{y} \in \mathbb{R}^{m} \) from a matrix \( \mathbf{A} \in \mathbb{R}^{m\times n} \) and a vector \( \mathbf{x} \in \mathbb{R}^{n} \):

\( \mathbf{y} = \mathbf{A}\mathbf{x} \)

The i-th entry is the dot product of row i of \( \mathbf{A} \) and \( \mathbf{x} \):

\( y_i = \sum_{j=1}^{n} a_{ij}\,x_j \quad \text{for } i=1,\dots,m \)

Time complexity is \(O(mn)\). Values are computed in double precision and displayed with the selected rounding.

1. Set the number of rows \(m\) and columns \(n\). Vector length is \(n\).
2. Enter values directly in the grids, or paste from the text areas.
3. Click Calculate to compute \( \mathbf{y} = \mathbf{A}\mathbf{x} \).
4. Toggle Show steps to see each row’s dot product expansion.
5. Export results via CSV or PDF, or copy steps for reports.
6. Use Shareable URL to preserve inputs for later or teammates.

• Empty cells are treated as zero. Use precision control for rounding only the display.
• Local storage remembers your latest session on this device.

• Linear systems step: Given \( \mathbf{A} \) and an iterate \( \mathbf{x} \), compute \( \mathbf{A}\mathbf{x} \) inside iterative solvers (CG, GMRES).
• Computer graphics: Apply linear transforms to coordinates, where \( \mathbf{x} \) contains point components.
• Dynamics & control: State update \( \mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{b} \).
• Markov chains: Next-state probabilities from transition matrix times distribution vector.
• Neural networks: Dense layer output \( \mathbf{y} = \mathbf{W}\mathbf{x} + \mathbf{b} \).
• Data pipelines: Feature mixing, filters, and dimensionality reduction steps.

Use this tool to prototype the multiply step before coding full pipelines.

• Complexity: \(O(mn)\) multiplies and adds; cost grows with size.
• Sparsity: Exploit zero entries by storing nonzeros only; compute faster and use less memory.
• Conditioning: Sensitivity obeys \( \frac{\lVert \Delta \mathbf{y}\rVert}{\lVert \mathbf{y}\rVert} \lesssim \kappa(\mathbf{A}) \frac{\lVert \Delta \mathbf{x}\rVert}{\lVert \mathbf{x}\rVert} \), where \( \kappa(\mathbf{A}) \) is the condition number.
• Scaling: Normalize inputs to mitigate overflow/underflow; prefer comparable magnitudes.
• Accumulation order: Summation order affects rounding error; Kahan summation reduces drift.
• Cache locality: Row-major access improves throughput for \( \mathbf{A}\mathbf{x} \) on typical CPUs.

Displayed precision affects formatting only; raw computations keep full double precision internally.