The matrix denotation. Clarification on Shear?

Question

I am given a Matrix [3 5; 5 -3]

I am told to describe all the possible denotations of the matrix. These choices are:

a) Rotation with Scaling

b) Rotation without Scaling

c) Reflection without Scaling

d) Reflection with scaling

e) vertical shear

f) Horizontal Shear

From a quick glance, formulaically, the matrix only falls under the category of reflection without scaling. However, I am uncertain of how a matrix denoting vertical shear or horizontal shear would look like?

Answer 1

Note that

• $A(1,0)=(3,5)$

• $A(0,1)=(5,-3)$

and $\det(A)=34$ therefore it shoul be a reflection with scaling. To check that completely we can find eigenvalue and eigenvectors.

We can exclude shear matrix since in that case angles are preserved, indeed $A^TA=34^2I$ and up to the scaling the transformation is an isometry.