Positive Definite Matrix - I think this is correct? Just want reassurance.

[Chronothread]

Homework Statement

For what b is the following matrix positive definite?
($$\stackrel{1}{b}$$ $$\stackrel{b}{4}$$)

(Sorry for the strange way I have that matrix represented)

Homework Equations

For a positive definite matrix
x$$^{T}$$Ax
for all nonzero x in $$\Re^n$$

The Attempt at a Solution

x$$^{T}$$Ax = x$$^{2}_{1}$$+2bx$$_{1}$$x$$_{2}$$+4x$$^{2}_{2}$$

And the only time this is a strictly positive number for all x is when b is zero correct?
Thanks!

(Also, if you have suggestions on how I can better represent my math using tex then I am doing please let me know)

 

[Dick]

No, there are other values of b that make that positive definite. Complete the square.

 

[Chronothread]

Ah yes, thank you.

Just to check quick then I get

(x$$_{1}$$+bx$$_{2}$$)$$^{2}$$+(4-b$$^{2}$$)x$$^{2}_{2}$$

Which is positive when -2$$\leq$$ b $$\leq$$ 2, correct?

Thanks again.

 

[statdad]

Remember that if $$A$$ is an $$n \times n$$ positive definite matrix then, among other things, the upper left element of the matrix must be positive, the determinant of the upper left $$2 \times 2$$ submatrix must have a positive determinant, and so on. For your matrix to be positive definite this means the upper left entry must be positive (check) and the determinant of the matrix has to be positive. What does that fact tell you about $$b$$ ? (Just an alternate way of considering this type of problem)

 

[Dick]

Ah yes, thank you.

Just to check quick then I get

(x$$_{1}$$+bx$$_{2}$$)$$^{2}$$+(4-b$$^{2}$$)x$$^{2}_{2}$$

Which is positive when -2$$\leq$$ b $$\leq$$ 2, correct?

Thanks again.

I wouldn't say b=2 or b=(-2) is going to give you a positive definite matrix.