Visualizing Matrix Transforms

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Often when linear algebra and matrices are introduced, the matrix operations are explained, but minimal information is presented to provide intuition into what what to expect from the product of a matrix and a vector other than a new vector.

For a symmetric matrix (M), we can generate a scalar quadratic function F by the formula F=xT*M*x. One fascinating result that help us visualize the matrix transformation is that the production Mx is equal to 1/2 the gradient of F. The gradient vector is oriented normal to the constant surfaces of the function F. The closer the constant surfaces are together, the longer the gradient vector. Note that this approach is valid for symmetric matrices of n rows and columns, but visualization is only possible in 2 and 3 dimensions.

This activity shows the constant surfaces of the function F=xT*M*x. It shows that when the result vector b of Mx=b is placed at the tip of the vector x, that this vector is normal to the constant surfaces of F. When the eigenvectors are both positive or both negative, the function F is an ellipse, and the eigenvectors are oriented along the principle axes of the ellipse (along the principle axes, the normal to the constant surfaces is along the principle axis). When one eigenvector is negtive and the other positive, the function F is a hyperbola. Move the vector x around to see how the transformation behaves (notice that it is consistent with the gradient of F).

View the GeoGebra Worksheet.

For antisymmetric matrices (A), F = xT*A*x = 0 and this approach doesn't work directly. However the approach does work for A*A which is symmetric. This visualization doesn't address antisymmetric matrices however.