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)