I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,

H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\alpha$ is defined as, $0<\alpha < \frac{2}{trace(A^{T} A)}$.

Can I claim that each one of therm in my matrix H will be less than or equal to 1?

I believe that the solution is yes, since I have not been able to come up with the a single scenario where this claim is not true.

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,

H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\alpha$ is defined as, $0<\alpha < \frac{2}{trace(A^{T} A)}$.

Can I claim that each one of therm in my matrix H will be less than or equal to 1?

I believe that the solution is yes, since I have not been able to come up with the a single scenario where this claim is not true. But I am not able to prove this.