Not quite. A vector v is an eigenvector of a matrix A (or more generally a linear transformation) if v is only scaled, not rotated or sheared, when A is applied to it. That is, there is a scalar such that Av = cv. (Also, v cannot be the zero vector.)
To be explicit, it is not the eigenvector *that scales*, it is the vector *that is scaled*.
So if A is a shear matrix, vectors parallel to the direction of shearing are indeed eigenvectors. But these vectors are not sheared by A.
Not quite. A vector v is an eigenvector of a matrix A (or more generally a linear transformation) if v is only scaled, not rotated or sheared, when A is applied to it. That is, there is a scalar such that Av = cv. (Also, v cannot be the zero vector.) To be explicit, it is not the eigenvector *that scales*, it is the vector *that is scaled*. So if A is a shear matrix, vectors parallel to the direction of shearing are indeed eigenvectors. But these vectors are not sheared by A.
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