Inner Product Spaces of 2x2 Matrix

[abbasb]

Homework Statement

Show that <U,V> = u1.v1 + u2.v3 + u2.v3 + u4.v4 is NOT an inner product on M2x2

Homework Equations

U: row 1 = [u1 u2] row 2 = [u3 u4] V: row 1 = [v1 v2] row 2 = [v3 v4]

The Attempt at a Solution

As I went through each of the axioms, I found that they were all correct (at least the way I did it), I've been stuck for quite some time now. I found that the symmetry axiom, additivity axiom and homogeneity axiom all remained true. So I'm assuming the positivity axiom doesn't hold true, I'm just confused as how to show this.

 

[Mark44]

Homework Statement

Show that <U,V> = u1.v1 + u2.v3 + u2.v3 + u4.v4 is NOT an inner product on M2x2

Your formula looks like it might have a typo, with u2.v3 appearing twice. Is that how the problem is worded?

Homework Equations

U: row 1 = [u1 u2] row 2 = [u3 u4] V: row 1 = [v1 v2] row 2 = [v3 v4]

The Attempt at a Solution

As I went through each of the axioms, I found that they were all correct (at least the way I did it), I've been stuck for quite some time now. I found that the symmetry axiom, additivity axiom and homogeneity axiom all remained true. So I'm assuming the positivity axiom doesn't hold true, I'm just confused as how to show this.

 

[abbasb]

Sorry, its supposed to be <U,V> = u1.v1 + u2.v3 + u3.v2 + u4.v4

and U and V are supposed to be matrices...couldn't find a better way to show them as so.

 

[Mark44]

I'm leaning toward the positivity thing, too. Can you come up with a 2x2 matrix U for which <U, U> = u₁² + u₂u₃ + u₃u₂ + u₄² < 0? Try playing around with a mix of positive and negative numbers for which the sume of u₂u₃ and u₃u₂ is more negative than the two squared terms are positive.

 

[abbasb]

I'm leaning toward the positivity thing, too. Can you come up with a 2x2 matrix U for which <U, U> = u₁² + u₂u₃ + u₃u₂ + u₂² < 0? Try playing around with a mix of positive and negative numbers for which the sume of u₂u₃ and u₃u₂ is more negative than the two squared terms are positive.

But shouldn't I be looking for a 2x2 matrix such that <U,U> = u1^2 + 2(u2.u3) + u4^2 < 0? And the only way I could find such a matrix is through trial and error?

 

[Mark44]

But shouldn't I be looking for a 2x2 matrix such that <U,U> = u1^2 + 2(u2.u3) + u4^2 < 0? And the only way I could find such a matrix is through trial and error?

First question: yes. u¹₂ and u₄² are always going to be nonnegative, so can you fiddle with values of u₂ and u₃ so that 2u₂u₃ is more negative than the two squared terms are positive?
Second question: Yes, but using judicious trial and error. You're only dealing with 2x2 matrices, and you can start with 1s in the main diagonal positions. That leaves you only two other positions to fill in.

BTW, I edited my previous post to fix a subscript.