# Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $$N \times N$$ to that of another DCT dictionary of size $$M \times M$$, where $$M>N$$. I always found that the highest value of $$DCT_{M \times M} > DCT_{N \times N}$$? Is there any mathematical reason behind it? Can this be generalized for all orthonormal dictionaries?

• It might help if you say what "orthonormal dictionary " means, and what its "value" is. These are not standard terms. – kimchi lover Feb 17 at 13:35
• I meant, the orthogonality property should be maintained. So, $AA^T=A^TA=I$. A good example might be discrete cosine transform, Gabor frame etc. – God_Help Feb 18 at 6:12