Hermitian Matrices |
Given a matrix A of dimension m ´ k (where m denotes the number of rows and k denotes the number of columns) and a matrix B of dimension k ´ n, the matrix product AB is defined as the m ´ n matrix with the components |
for m ranging from 1 to m and for n ranging from 1 to n. Notice that matrix multiplication is not generally commutative, i.e., the product AB is not generally equal to the product BA. |
The transpose A^{T} of the matrix A is defined as the k ´ m matrix with the components |
for m ranging from 1 to m and for k ranging from 1 to k. Notice that transposition is distributive, i.e., we have (A+B)^{T} = (A^{T} + B^{T}). |
Combining the preceding definitions, the transpose of the matrix product AB has the components |
Hence we've shown that |
We can also define the complex conjugate A^{*} of the matrix A as the m ´ k matrix with the components |
Notice that the matrix A can be written as the sum A_{R} + iA_{I} where A_{R} and A_{I} are real valued matrices. The complex conjugate of A can then be written in the form |
We also note that transposition and complex conjugation are commutative, i.e., we have (A^{T})^{*} = (A^{*})^{T}. Hence the composition of these two operations (in either order) gives the same result, called the Hermitian conjugate (named for the French mathematician Charles Hermite, 1822-1901) and denoted by A^{H}. We can express the components of A^{H} as follows |
A Hermitian matrix is defined as a matrix that is equal to its Hermitian conjugate. In other words, the matrix A is Hermitian if and only if A = A^{H}. Obviously a Hermitian matrix must be square, i.e., it must have dimension m ´ m for some integer m. |
The Hermitian conjugate of a general matrix product satisfies an identity similar to (1). To prove this, we begin by writing the product AB in the form |
Thus the Hermitian conjugate can be written as |
Applying identity (1) to the transposed products, we have |
We recognize the right hand side as the product of the Hermitian conjugates of B and A, i.e., |
Consequently we have the identity |
We're now in a position to prove an interesting property of Hermitian matrices, namely, that their eigenvalues are necessarily real. Recall that the scalar l is an eigenvalue of the (square) matrix A if and only if there is a column vector X such that |
Taking the Hermitian conjugate of both sides, applying identity (2), and noting that multiplication by a scalar is commutative, we have |
Now, if A is Hermitian, we have (by definition) A^{H} = A, so this becomes |
If we multiply X by both sides of this equation, and if we multiply both sides of the original eigenvalue equation by XH, we get |
Since the left hand sides are equal, and since multiplication by a scalar is commutative, we have l = l^{*}, and therefore l is purely real. |
Of course, the converse is not true. A matrix with real eigenvalues is not necessarily Hermitian. This is easily seen by examining the general 2 ´ 2 matrix |
The roots of the characteristic polynomial |
are |
The necessary and sufficient condition for the roots to be purely real is that both of the following relations are satisfied |
If the matrix is Hermitian we have a = d = 0, b = c, and b = -g, in which case the left hand expression reduces to a sum of squares (necessarily non-negative) and the right hand expression vanishes. However, it is also possible for these two relations to be satisfied even if the original matrix is not Hermitian. For example, the matrix |
is not Hermitian if r ¹ s, but it has the eigenvalues |
which are purely real provided only that rs ³ 0. |
Returning to Hermitian matrices, we can also show that they possess another very interesting property, namely, that their eigenvectors are mutually orthogonal (assuming distinct eigenvalues) in a sense to be defined below. To prove this, let l_{1} and l_{2} denote two distinct eigenvalues of the Hermitian matrix A with the corresponding eigenvectors X_{1} and X_{2}. (These subscripts signify vector designations, not component indices.) Then we have |
Taking the Hermitian conjugate of both sides of the left hand equation, replacing A^{H} with A, noting that l_{1}^{*} = l_{1}, and multiplying X_{2} by both sides gives |
Now we multiply both sides of the right hand equation by X_{1}^{H} to give |
The left hand sides of these last two equations are identical, so subtracting one from the other gives |
Since the eigenvalues are distinct, this implies |
which shows that the "dot product" of X_{2} with the complex conjugate of X_{1} vanishes. In general this inner product can be applied to arbitrary vectors, and we sometimes use the bra/ket notation introduced by Paul Dirac |
where, as always, the asterisk superscript signifies the complex conjugate. (The subscripts denote component indices.) Terms of this form are a suitable "squared norm" for the same reason that the squared norm of an individual complex number z = a + bi is not z^{2}, but rather z*z = a^{2} + b^{2}. |
Hermitian matrices have found an important application in modern physics, as the representations of measurement operators in Heisenberg's version of quantum mechanics. To each observable parameter of a physical system there corresponds a Hermitian matrix whose eigenvalues are the possible values that can result from a measurement of that parameter, and whose eigenvectors are the corresponding states of the system following a measurement. Since a measurement yields precisely one real value and leaves the system in precisely one of a set of mutually exclusive (i.e., "orthogonal") states, it's natural that the measurement values should correspond to the eigenvalues of Hermitian matrices, and the resulting states should be the eigenvectors. |