Unitary Operation On A Complex Matrix

In summary, the conversation is discussing the notation and definition of ##\hat A## acting on a complex matrix ##A##, where ##I_n## is the identity matrix. There is some confusion about the notation and its meaning, but it is unclear without more context.
  • #1
Bashyboy
1,421
5
Hello everyone,

Let ##A = (\alpha_{ij})## be an $n \times n# complex matrix. Define ##\hat## acting on ##A## as producing the matrix ##\hat{A} = (\alpha_{ij} I_n)##.

I don't understand what this is saying. Isn't ##I_n## the identity matrix, and therefore the product of it with any matrix results in the other matrix? If this is so, wouldn't ##\hat{A} = (\alpha_{ij} I_n) = (\alpha_{ij}) = A##?
 
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  • #2
Without seeing more of the context, I would think that ##\hat{A}## is an n2 by n2 matrix where every element, ##\alpha_{ij}##, of A is replaced by the n by n submatrix ##\alpha_{ij}I_n##
 
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  • #3
Bashyboy said:
Hello everyone,

Let ##A = (\alpha_{ij})## be an ##n \times n## complex matrix. Define ##\hat{}## acting on ##A## as producing the matrix ##\hat{A} = (\alpha_{ij} I_n)##.

I don't understand what this is saying. Isn't ##I_n## the identity matrix, and therefore the product of it with any matrix results in the other matrix? If this is so, wouldn't ##\hat{A} = (\alpha_{ij} I_n) = (\alpha_{ij}) = A##?
##\alpha_{ij}## isn't a matrix.

The notation in the sentence you're asking about is very strange. If ##(\alpha_{ij})## denotes the matrix with ##\alpha_{ij}## on row i, column j, then ##(\alpha_{ij}I_n)## should denote the matrix with ##\alpha_{ij}I_n## on row i, column j. But this is a matrix whose every component is a matrix. Is it possible that this is what they meant? If yes, then I agree with FactChecker. If no, then maybe they meant ##\hat A=(\alpha_{ij}(I_n)_{ij}) =(\alpha_{ij}\delta_{ij})##.
 
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1. What is a unitary operation on a complex matrix?

A unitary operation on a complex matrix is a type of linear transformation that preserves the length of a vector. In other words, it maintains the norm of a vector, which is the square root of the sum of the squares of its components.

2. How is a unitary operation different from a normal matrix operation?

A unitary operation is different from a normal matrix operation in that it preserves the norm of a vector, whereas a normal matrix operation can change the norm of a vector. Additionally, a unitary operation is always reversible, while a normal matrix operation may not be.

3. What is the significance of unitary operations on complex matrices?

Unitary operations on complex matrices are significant because they have many important applications in mathematics, physics, and engineering. For example, they are used in quantum mechanics to represent physical transformations and in signal processing for data compression and encryption.

4. How are unitary operations related to Hermitian matrices?

Unitary operations are closely related to Hermitian matrices because a matrix is unitary if and only if its inverse is equal to its conjugate transpose, which is a property of Hermitian matrices. In other words, a unitary operation on a complex matrix is a Hermitian transformation.

5. Can any complex matrix be transformed into a unitary matrix?

No, not every complex matrix can be transformed into a unitary matrix. In order for a matrix to be unitary, it must satisfy certain conditions, such as having unit length columns and being square. If a matrix does not meet these criteria, it cannot be transformed into a unitary matrix.

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