Proving an Identity Involving Matrices and Inner Products

In summary, an identity involving matrices and inner products is a mathematical statement that equates two expressions involving these operations. Proving these identities is important for verifying their validity and understanding relationships between matrices and inner products. The steps to proving these identities involve breaking down the expressions, using properties and manipulation, and common techniques include algebraic manipulations, properties, and substitution. While not all identities can be proven, if valid, they can always be proven using the right techniques.
  • #1
devious_
312
3
Is there a non-ugly proof of the following identity:
[tex]\langle Ax,y \rangle = \langle x,A^*y \rangle[/tex]
where A is an nxn matrix over, say, [itex]\mathbb{C}[/itex], A* is its conjugate transpose, and [itex]\langle \cdot , \cdot \rangle[/itex] is the standard inner product on [itex]\mathbb{C} ^n[/itex].
 
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  • #2
Just start writing out the left side using [tex]<a,b>=a^\dagger b[/tex].
 
  • #3
Thanks.

Although using my definition for the standard inner product I had to use [itex] \langle a , b \rangle = a^t \bar{y}[/itex], but it all worked out in the end.
 

Related to Proving an Identity Involving Matrices and Inner Products

1. What is an identity involving matrices and inner products?

An identity involving matrices and inner products is a mathematical statement that equates two expressions involving matrices and inner products, which are mathematical operations that involve multiplying and adding vectors and matrices.

2. Why is it important to prove an identity involving matrices and inner products?

Proving an identity involving matrices and inner products is important because it allows us to verify the validity of a mathematical statement and demonstrate the relationships between different matrices and inner products.

3. What are the steps to proving an identity involving matrices and inner products?

The steps to proving an identity involving matrices and inner products usually involve breaking down the expressions into smaller parts, using properties of matrices and inner products, and manipulating the expressions until they are equivalent.

4. What techniques can be used to prove an identity involving matrices and inner products?

Common techniques used to prove an identity involving matrices and inner products include using algebraic manipulations, applying properties of matrices and inner products, and using strategies such as substitution and factoring.

5. Can an identity involving matrices and inner products always be proven?

Not necessarily. There may be some identities that are not true for all matrices and inner products, or that require advanced mathematical techniques to prove. However, if a valid identity exists, it can always be proven using the appropriate techniques.

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