Showing determinant of product is product of dets for linear operators

In summary, the problem is to show that the determinant of the product of two normal linear operators, A and B, is equal to the product of their determinants. This can be proven by choosing a convenient orthonormal basis, such as the standard cartesian basis, where each operator can be represented by the identity matrix. From this representation, it is straightforward to show the desired identity.
  • #1
Hakkinen
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Homework Statement


Assume A and B are normal linear operators [itex][A,A^{t}]=0[/itex] (where A^t is the adjoint)

show that det AB = detAdetB


Homework Equations





The Attempt at a Solution


Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets. I'm not really sure how to proceed though so any assistance would be appreciated.
 
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  • #2
I think I've got a solution for this but I'm not sure if it's general enough.

So since each operators eigenbasis forms an orthonormal set you can choose any convenient orthonormal basis. If I pick the standard cartesian basis (i,j,k) then each operator can be represented by the identity matrix. From here it is straight forward to show the identity.
 

1. What is the determinant of a product of linear operators?

The determinant of a product of linear operators is equal to the product of the determinants of each individual operator.

2. Why is the determinant of a product of linear operators equal to the product of their determinants?

This property is a result of the distributive and associative properties of matrix multiplication. It can be proven using the definition of the determinant as the sum of products of elements in a matrix.

3. Does this property hold for non-square matrices?

No, this property only holds for square matrices. For non-square matrices, the product of their determinants is not defined.

4. Can this property be extended to a product of more than two linear operators?

Yes, this property can be extended to a product of any number of linear operators. The determinant of the product is equal to the product of the determinants of each individual operator.

5. How is this property useful in linear algebra?

This property is useful in simplifying computations involving determinants of linear operators. It also helps in understanding the relationship between the determinants of linear operators and their composition.

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