Applications of the diagonal representation I
When is symmetric, we can use the representation
1. Matrix positivity in terms of eigenvalues
Exercise 1. Let be a symmetric matrix. It is positive (non-negative) if and only if its eigenvalues are positive (non-negative).
Proof. By definition, we have to consider the quadratic form Denoting we know that an orthogonal matrix preserves the norm: This allows us to obtain a lower bound for the quadratic form
This implies the statement.
2. Analytical functions of a matrix
Definition 1. For a square matrix all non-negative integer powers are defined. This allows us to define an analytical function of a matrix whenever a function has the Taylor decomposition
Example 1. The exponential function has the decomposition Hence, Differentiating this gives
( the constant term disappears)
This means that solves the differential equation To satisfy the initial condition instead of we can consider This matrix function solves the initial value problem
Calculating all powers of a matrix can be a time-consuming business. The process is facilitated by the knowledge of the diagonal representation.
Exercise 2. If is symmetric, then for all non-negative integer Hence,
Proof. The equation
shows that to square it is enough to square In a similar fashion we can find all non-negative integer powers of
3. Linear differential equation
Example 1 about the exponential matrix function involves a bit of guessing. With the diagonal representation at hand, we can obtain the same result in a more logical way.
One-dimensional case. is equivalent to Upon integration this gives
General case. In case of a matrix the first equation of (2) is The fact that the x's on the right are mixed up makes the direct solution difficult. The idea is to split the system and separate the x's.
Premultiplying (2) by we have or, denoting and using (1), The last system is a collection of one-dimensional equations Let be the initial vector. From (4) In matrix form this amounts to Hence, as in Exercise 2,
This is the same solution obtained above. The difference is that here we assume symmetry and, as a consequence, can use Example 2.