Applications in Engineering
Positive-definite matrices have some nice properties which we will
exploit in finding more convenient decompositions which require less
Useful background for this topic includes:
- Matrix multiplication and inner (dot) products
A symmetric n × n matrix M is said to be
positive definite if for all nonzero vectors v, the
product vTMv > 0.
Looking more closely at this product, we see that it
is the dot product of vT and Mv.
You will recall that the dot product of two vectors is equal to
vTMv = ||v||2 ||Mv||2 cos(θ),
where θ is the angle between v and Mv, and
and thus, for the dot product to be positive, it means that the image of
v must have an angle θ less than 90o, i.e., |θ| < π/2.
It is unreasonable to test every possible vector v to determine
if a matrix is positive definite, however, there are some cases which appear
quite often in engineering where it is quite simple to state that a matrix
is positive definite. These are covered on the howtos. The following
are some interesting theorems related to positive definite matrices:
A matrix is invertible if and only if all of the eigenvalues are non-zero.
Proof: Please refer to your linear algebra text.
A positive definite matrix M is invertible.
Proof: if it was not, then there must be a non-zero vector
x such that Mx = 0. Therefore
xTMx = 0 which contradicts our assumption about
M being positive definite.
All the eigenvalues with corresponding real eigenvectors of a positive
definite matrix M are positive.
Proof: if x is an eigenvector of M then Mx = λx
and therefore xTMx = λ||x||2. For
this to be positive, it follows that λ > 0.
A symmetric matrix is positive definite if:
- all the diagonal entries are positive, and
- each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column.
An arbitrary symmetric matrix is positive definite if and only if
each of its principal submatrices
has a positive determinant. This is known as Sylvester's criterion.
Note that only the last case does the implication go both ways.
A matrix M is row diagonally dominant if
for i = 1, ..., n, column diagonally dominant if
for i = 1, ..., n, and diagonally dominant if it is both row and column diagonally dominant.
Thus, we may restate our first definition as:
A symmetric row diagonally-dominant matrix with positive diagonal entries is positive definite.
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Show that if A is invertible, then ATA is
If A is invertible, then Av ≠ 0 for any vector v ≠ 0.
Therefore vT(ATA)v =
(vTAT)(Av) which is the vector
Av dotted with itself, that is, the square of the norm (or length) of the
vector. As Av ≠ 0, the norm must be positive, and therefore
vT(ATA)v > 0.
Create a random positive-definite n × n matrix in which is also diagonally dominant.
( rand(n, n) + (n - 1)*eye( n ) )
Which of the following matrices are positive definite?
Applications to Engineering
The conductance matrix of a RLC circuit is positive definite.
Additionally, we will see that the matrix defined when performing
least-squares fitting is also positive definite.
As an alternate example, the Hurwitz criteria for the stability
of a differential equation requires that the constructed matrix
be positive definite. This is seen in signals and linear systems.
A test for positive definiteness requires that the matrix is symmetric and
that all the eigenvalues are positive.
all( all( M == M' ) ) & min( eig( M ) ) > 0
Alternatively, one may use the test
M = [...]; % assume M is square
isposdef = true;
if ( det( M(1:i, 1:i) ) <= 0 )
isposdef = false;
isposdef % 0 if false, 1 if true
The following commands in Maple tests if a matrix is positive definite:
M := <<-6, 12, 2.4, 0>|<-1, 2, 10.4, 1>|<3.25, 1, -1.8, 14.8>|<10.25, 0, 2, 1.2>>;
LinearAlgebra[IsDefinite]( M );
For more help on either of these routines or on the LinearAlgebra