In this chapter, the linear transformations are from a given finite dimensional vector space to itself. Observe that in this case, the matrix of the linear transformation is a square matrix. So, in this chapter, all the matrices are square matrices and a vector means for some positive integer
To solve this, consider the Lagrangian
Partially differentiating with respect to for we get
and so on, till
Therefore, to get the points of extrema, we solve for
We therefore need to find a and such that for the extremal problem.
be a matrix of order
In general, we ask the question:
For what values of there exist a non-zero vector such that
By Theorem 2.5.1, this system of linear equations has a non-zero solution, if
So, to solve (6.1.4), we are forced to choose those values of for which Observe that is a polynomial in of degree We are therefore lead to the following definition.
Some books use the term EIGENVALUE in place of characteristic value.
has a non-zero solution. height6pt width 6pt depth 0pt
Consider the matrix Then the characteristic polynomial of is
Given the matrix recall the linear transformation defined by
Suppose is a root of the characteristic equation Then is singular and Suppose Then by Corollary 4.3.9, the linear system has linearly independent solutions. That is, has linearly independent eigenvectors corresponding to the eigenvalue whenever
In general, if are linearly independent vectors in then are eigenpairs for the identity matrix,
[Hint: Recall that if the matrices and are similar, then there exists a non-singular matrix such that ]
So, by definition of trace.
Hence, we get the required result. height6pt width 6pt depth 0pt
Let be an matrix. Then in the proof of the above theorem, we observed that the characteristic equation is a polynomial equation of degree in Also, for some numbers it has the form
Note that, in the expression is an element of Thus, we can only substitute by elements of
It turns out that the expression
holds true as a matrix identity. This is a celebrated theorem called the Cayley Hamilton Theorem. We state this theorem without proof and give some implications.
holds true as a matrix identity.
Some of the implications of Cayley Hamilton Theorem are as follows.
That is, we just need to compute the powers of till
In the language of graph theory, it says the following:
``Let be a graph on vertices. Suppose there is no path of length or less from a vertex to a vertex of Then there is no path from to of any length. That is, the graph is disconnected and and are in different components."
This matrix identity can be used to calculate the inverse.
Let the result be true for We prove the result for We consider the equation
This is an equation in eigenvectors. So, by the induction hypothesis, we have
But the eigenvalues are distinct implies for We therefore get for Also, and therefore (6.1.9) gives
Thus, we have the required result. height6pt width 6pt depth 0pt
We are thus lead to the following important corollary.
In each case, what can you say about the eigenvectors?
A K Lal 2007-09-12