Matrix-Vector Multiplication Visualizer
Enter a $2\times 2$ matrix $A$ and a vector $\mathbf{x}$ to visualize the product $A\mathbf{x}$. Explore eigenvectors, eigenvalues, and geometric interpretations.
A $2\times 2$ matrix $A$ defines a linear transformation that maps every vector $\mathbf{x}$ in the plane to a new vector $A\mathbf{x}$. This tool shows that multiplication geometrically.
- Blue arrow – the input vector $\mathbf{x}$
- Red arrow – the output vector $A\mathbf{x}$
- Green / orange dashed lines – eigenvectors of $A$ (when the eigenvalues are real)
An eigenvector $\mathbf{v}$ satisfies $A\mathbf{v} = \lambda\mathbf{v}$: the matrix only scales it by the factor $\lambda$, never changing its direction (though it may reverse direction when $\lambda < 0$). The scalar $\lambda$ is called the eigenvalue.
When $A\mathbf{x}$ is parallel to $\mathbf{x}$, the input vector lies along an eigenvector direction — the canvas will display $A\mathbf{x} = \lambda\mathbf{x}$. For example, a $180°$ rotation has $A = -I$, so every vector is an eigenvector with $\lambda = -1$.
When the eigenvalues are complex (as in most rotations other than $0°$ and $180°$), no real eigenvectors exist — the matrix rotates every vector off its original direction. The tool will display "Complex eigenvalues: $\lambda = a \pm bi$" in this case.
Use the preset examples to explore rotations, reflections, shears, projections, and matrices with specific integer eigenvalues.