Showing something satisfies Inner Product - Involves Orthogonal Matrices

In summary, the conversation discusses the task of proving that the product <x, y> A = x · Ay is an inner product for R3. The conversation mentions using the properties of Z-1 and ZT, as well as the fact that (ATx)⋅x = x ⋅ Ax. The conversation also provides the hint that (Zx)^T = x^T Z^T and that D is all positive, leading to the conclusion that Z^T D Z should also be non-negative.
  • #1
Circular_Block
2
0

Homework Statement


[/B]
Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal.
Show that the product <x, y> A = x · Ay is an inner product for R3.

Homework Equations


None

The Attempt at a Solution



I've shown that x · Dy is an inner product. I know that Z-1 is equal to ZT. I believe that will lead me somewhere. I'm just having trouble showing the property <x, x> ≥ 0. I also know that (ATx)⋅x = x ⋅ Ax.

Just missing one step. I don't know what it is.
 
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  • #2
Hint: ## (Zx)^T = x^T Z^T ##
 
  • #3
Doing that you'll end up with (ATx)⋅x right?
 
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  • #4
Circular_Block said:
Doing that you'll end up with (ATx)⋅x right?

x.y=x^Ty. That turns a dot product into a matrix product. Add that to the list of clues.
 
  • #5
Don't neglect that D is all positive. So ##Z^T D Z## should also be non-negative, right?
 

FAQ: Showing something satisfies Inner Product - Involves Orthogonal Matrices

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a scalar value. It is often denoted by <u,v> and satisfies properties such as linearity, symmetry, and positive definiteness.

2. What are orthogonal matrices?

Orthogonal matrices are square matrices that have a special property of being orthogonal, meaning that their columns and rows are all orthogonal to each other. This means that when multiplied by their transpose, they result in the identity matrix.

3. How do you show that something satisfies an inner product?

To show that something satisfies an inner product, you must demonstrate that it satisfies the properties of linearity, symmetry, and positive definiteness. This can be done by performing algebraic manipulations and using mathematical definitions.

4. What is the significance of orthogonal matrices in inner product?

Orthogonal matrices are important in inner product because they preserve the length and angles of vectors when they are multiplied by them. This allows for the calculation of inner products without changing the geometric properties of the vectors.

5. Can any matrix satisfy an inner product?

No, not all matrices can satisfy an inner product. For a matrix to satisfy an inner product, it must have the properties of linearity, symmetry, and positive definiteness. Orthogonal matrices are a special case that satisfy these properties, but not all matrices do.

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