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Let be column vectors in and let be the corresponding real matrix. Then the following inequality holds:

(1)

where is the Euclidean norm on vectors in . In continuity , give the geometrical interpretation of the inequality above.

Solution

By the Gramm – Schmidt process we can establish the existence of an orthonormal basis such that

(2)

for each . Now, we may write for the corresponding real and orthogonal matrix. By orthogonality each vector in has an expansion as:

On the other hand implies that each vector has a shorter expansion of the form:

(3)

Alternatively let be the upper triangular matrix defined as:

Then is restated as and using again the fact that has orthonormal columns and the fact that is upper triangular we get:

Notes:

• The above argument also shows that there exists equality if and only if

for each . That is , if and only if, . This can only be achieved if the vectors are pairwise orthogonal.

• The geometrical interpretation of this inequality is the following: The volume of an dimensional parallelepiped produced by vectors can not exceed the product of their measures.