Hermitian Matrix

Algebra Level 4

If we view the set of all Hermitian matrices of order n n as R \mathbb R -vector space, what is its dimension?

Definition: A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.

n ( n + 1 ) n(n+1) n 2 2 \frac{n^2}{2} n ( n + 1 ) 2 \frac{n(n+1)}{2} n 2 n^2

Brian Lie by Brian Lie , 26, China

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1 solution

Chris Lewis
Jul 15, 2019

It's pretty easy to check the axioms of a vector space are satisfied by the set of all Hermitian matrices under matrix addition and under scalar multiplication by real numbers.

The entries of a Hermitian matrix are fully determined by the entries on the main diagonal (which must be real) and the entries above the main diagonal (which can be complex). There are n n entries on the main diagonal, and n ( n − 1 ) 2 \frac{n(n-1)}{2} entries above it. Considering the real and imaginary parts of the entries above the main diagonal, each entry contributes two dimensions, for a total of n + n ( n − 1 ) = n 2 n+n(n-1)=\boxed{n^2} dimensions overall.

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