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I need some visual intuition behind what exactly a symmetric matrix transformation does. In a $2 \times 2$ and $3 \times 3$ vector space, what are they generally?

JobHunter69
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  • Just look at the image of the unit circle. – user251257 Apr 07 '18 at 23:05
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    Adding to the comment above - the image of the unit circle under the transformation by the symmetric matrix is an ellipse, whose axes point in the directions of the eigenvectors of the symmetric matrix. – joshuaronis Jun 24 '19 at 22:11

1 Answers1

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A real symmetric matrix is always orthogonally diagonalizable, meaning that there's a basis for $\mathbb R^n$ consisting of mutually perpendicular eigenvectors of the matrix. Thus you can understand multiplying a column vector by a symmetric matrix geometrically as:

  • Express the input vector in a different rectangular coordinate system that depends on the matrix.
  • Multiply each coordinate by some constant that depends on which axis in the new coordinate system it corresponds to -- that is, stretch, shrink or flip each axis independently of each other.
  • Express the result back in the original coordinate system.
hmakholm left over Monica
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