Saturday, January 28, 2012

Positive and Positive Definite Matrices

We continue to examine some more aspects of Hermitian matrices and some general properties of all matrices. We will stick in this exposition to simple 2 x 2 matrices, but understand everything treated can be generalized to larger array square matrices. Consider first the case of a positive matrix.

Here, a Hermitian matrix is positive if all eigenvalues > (=) to 0. Let's consider this example for which we are to check whether it's positive: M =

(1 -φ .........i)
(-i.........1 - φ)

Write out:

M = (1 - φ)^2 - (-i)(i) = (1 - φ) (1 - φ) - (1) = φ^2 - 2φ + 1 - 1 = 0

Or:

φ^2 - 2φ = 0 so: φ( φ - 2) = 0

whence: φ1 = 0 and φ2 = 2

Thus, the condition is met that the eigenvalues (φ1, φ2) are equal to or greater than 0 so the matrix is positive.

We now consider any general symmetric matrix, i.e. such that A = t^A in the context of when such a matrix is positive definite. The conditions for this are:

i) All the eigenvalues are positive (e.g. φi > 0)

ii) All the determinants are positive (e.g. ac - bd > 0)

iii)The pivots ([ac - bd]/ a) > 0

iv) t^x A x > 0

Example: State whether matrix W =

(2.....6)
(6......19)

is positive definite.

Solution: We check each of the conditions (i - iv) to see if they are met.

First, the eigenvalues must be > 0. We check the calculations - see, e.g.
http://brane-space.blogspot.com/2011/12/revisiting-linear-algebra.html

to see : φ1 = 0.096 and φ2 = 20.904

so both meet the condition.

Check that the determinant D > 0:

D = (2) (19) - (6)(6) = 38 - 36 = 2

so the condition is met.

Check that the pivot P > 0.

P = ([ac - bd]/ a) = 2/ 2 = 1

so, the condition is met.

Check to see if: t^x A x > 0

We let: x =

[x1]
[x2]

so that: t^x = [x1 ...x2]

Then we obtain the operation: t^x W x =

[x1 ...x2] [2x1 + 6x2]
...............[6x1 + 19x2]

Yielding the quadratic form: 2x1^2 + 12x1 x2 + 19 x2^2

where: a = 2, 2b = 12 so b = 12/2 = 6 and lastly c = 19

Hence, it meets the condition since a, b, c are all positive.

Then the matrix is positive definite.


Problems:

Check each of the following matrices to determine if positive or positive definite or positive semi-definite (the case where D = ac - bd = 0)

1)

(cos π/2.........-sin π/2)
(sin π/2.........cos π/2 )

2)

(3.......1)
(0.......3)

3)

(2.....6)
(6.....18)


4)

(2i.....1)
(2.......i)

5)

(-½ i.....i)
(2i....... i)

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