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Definition and Applications of Hermitian Matrices
Amy Perrey -- Math 208
Abstract: A Hermitian matrix has eigenvalues and eigenvectors that conform to four basic
properties regarding their existence, type, linear independence, and relation to each other.
Real symmetric matrices, a subclass of Hermitian matrices, are defined as the set of matrices
for which the original matrix is equal to its transpose. (The elements of a real symmetric
matrix are in the set of real numbers)
The base definition of a Hermitian matrix is a matrix that is equal to its Hermitian
conjugate. The concept of the Hermitian conjugate was formulated by French mathematician
Charles Hermite in 1855 and is denoted by AH. The definition of a Hermitian conjugate is
based on the fact that transposition of a matrix and complex conjugation are commutative.
When dealing with real symmetric matrices, one may note that the Moore-Penrose
generalized inverse may be used to confirm that AT=A.
The eigenvalues and eigenvectors of a Hermitian matrix have four fundamental
properties. First, all eigenvalues are in the set of real numbers. Secondly, there exists a set of
n linearly independent eigenvectors, regardless of the order (multiple) of the eigenvalues
themselves. Third, given distinct eigenvalues, the eigenvectors form an orthogonal set.
Finally, it follows from the second and third property that when given an eigenvalue of
multiplicity m, it is possible to choose eigenvectors that are mutually orthogonal and linearly
independent.
Hermitian, or self-adjoint, matrices are largely used in applications of Heisenberg’s
quantum mechanics. A Hermitian matrix itself describes each observable parameter of a
physical system, while its eigenvalues describe possible values of measurements of that
parameter. The corresponding eigenvectors depict the system after a measurement is taken.
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Hermitian matrices are suitable because a real eigenvalue (measurement) corresponds to a set
of mutually exclusive (orthogonal) eigenvectors (states following a measurement).
For example, if vector a describes the position of the elements in a system, and b
describes the velocity of the elements in a system, then we may use the following example to
1
i 
describe the state of the system after the velocity is measured: Let a    and b    . If we
i 
1
form the matrix A = {a,b}, the eigenvalues (  1,  2) of the system are found by finding the
determinant of the matrix A-  I.
det(A
I )  (1 )(1 ) (i 2 )
  
1    i  
1  0
A  I  


i
1




2  2
 have been found, the corresponding eigenvalues may be found by
After the eigenvalues
substituting them backinto the original matrix A-  I and row reducing. We will demonstrate
with the eigenvalue  2 = 2 and the associated eigenvector v.
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 1  i  R



 i  1 R2 +iR1


1 i 0 
0 0 0 



x1  x 2i  0
x1  ix 2
i 
v 
 1
i 

If this procedure is repeated with the eigenvalue  1= 0, we find that the eigenvector u =   .
1
If u and v are orthogonal, the dot product of u andv will be equal to zero. Since i2 + 1 = 0,
 orthogonal, and each vector describes a unique state
we may conclude that vectors u and v are
of the system after measuring the respective eigenvalues.
References:
Boyce, William E., and Richard C. DiPrima. Elementary Differential Equations and Boundary
Value Problems. New York: John Wiley & Sons, Inc., 2001
Gan, Gaoxiong. On the relation between Moore's and Penrose's conditions. Int. J. Math.
Math. Sci. 30 (2002), no. 8, 505--509.
Weisstein, Eric W. "Hermitian Matrix." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/HermitianMatrix.html
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