Mathematical Methods (10/24.539)
III. Overview of Linear Algebra
Some Special Matrices
There are a number of matrices that deserve some special mention. In particular, of interest here are three classes of matrices -- Hermitian, Skew-Hermitian, and Unitary Matrices.
Hermitian matrices
satisfy the relationship
where the wavy line over an element implies that one should take the complex conjugate of that quantity. An example of a hermitian matrix is
Note that the diagonal elements of hermitian matrices must be real, since 
for the diagonal elements.
Skew-hermitian matrices
are similar to the above definition except for a negative relationship, or
In this case, the diagonal elements must be pure imaginary, since 
along the diagonal. This really says that 
, which clearly implies that 
. As an example, consider the following matrix,
A
unitary matrix is one that satisfies the expression
As an example, if 
is given by 
and 
should be the same as 
. As a check, let’s compute the product 
and see if it gives the identity matrix, or
Note that hermitian, skew-hermitian, and unitary matrices are, in general, complex matrices. However, a subset of each of these classes exists for the case of all real elements, and they go by the names
symmetric, skew-symmetric, and orthogonal, respectively.The nice thing about matrices within these classes is that we can characterize their eigenvalue spectrum, as follows:
1. The eigenvalues of a hermitian (or real symmetric) matrix are real.
2. The eigenvalues of a skew-hermitian (or real skew-symmetric) matrix are imaginary or zero.
3. The eigenvalues of a unitary matrix (or real orthogonal) matrix have absolute value of unity.
For the examples given above we can compute the eigenvalues, giving,
Hermitian matrix
Skew-hermitian matrix
Unitary matrix
where this last set of eigenvalues was obtained from Matlab. Also note that the magnitude of each eigenvalue for the unitary matrix is indeed unity. For example the magnitude of 
is given by
The form 
is a common combination of terms that occurs frequently in applications. In particular, if 
are both real, then the combination 
is referred to as a
is hermitian, then is real for any 
, and if 
is skew-hermitian, then 
is pure imaginary for any 
. These summary properties can be useful in some cases.
One final set of terminology related to the eigenvalues of a matrix still needs to be discussed – that is the concept of
similar matrices and related subjects. In particular, two matrices,
, are said to be similar if they satisfy the relation
where 
is a transformation matrix. This transformation is said to be a Similarity Transformation. The important point here is that similar matrices have the same eigenvalues. In addition, if is an eigenvector of , then 
is the eigenvector of 
corresponding to that same eigenvalue. We can show this by the following manipulations:
This set of expressions shows that 
is indeed an eigenvalue of 
and that 
is the eigenvector of that corresponds to eigenvalue 
.
If 
are distinct eigenvalues of an nxn matrix, then the corresponding eigenvectors 
form a linearly independent set and they represent a basis of eigenvectors in n dimensional space.
The modal matrix is a special matrix whose columns contain the linearly independent eigenvectors/basis vectors, or
Also we should note that any vector, 
, has a unique representation in n dimensional space simply as a linear combination of the basis vectors, or
Also note that a linear transformation, 
, in terms of the basis vectors, becomes
or
Now if we let the transformation matrix, , in the above similarity transformation expression [see eqn. (3.31)] be composed of the basis vectors for the n-dimensional problem, or
then,
which is a diagonal matrix with the eigenvalues of 
along the diagonal of 
. Also, note that a similar relationship that is often used is
A proof of the first relationship can be demonstrated as follows:
or 
and 
as given above.
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10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated August 1998).
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