Can I Write a Hermitian Matrix as A=A^(1/2)AH^(1/2)?

In summary, the conversation discusses whether a Hermitian matrix can be written as the product of its square root and its complex conjugate transpose. The participants clarify that the notation ##H/2## refers to the square root of the complex conjugation transpose. They also point out that since a Hermitian matrix is equal to its complex conjugate transpose, the original equation can be simplified to A = A^{1/2}A^{1/2}.
  • #1
EngWiPy
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Hello,

If I have a Hermitian matrix A, can I write it as:

[tex]\mathbf{A}=\mathbf{A}^{1/2}\mathbf{A}^{H/2}[/tex]

where superscript H denotes hermitian operation?

Thanks
 
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  • #2
What do you mean with ##H/2##? Do you mean

[tex](A^H)^{1/2}[/tex]

or something similar?
 
  • #3
micromass said:
What do you mean with ##H/2##? Do you mean

[tex](A^H)^{1/2}[/tex]

or something similar?

Yes exactly, it means the square root of the complex conjugation transpose of A.
 
  • #4
OK, but if ##A## is hermitian, then ##A^H = A##, no? So you can write your original post as

[tex]A = A^{1/2}A^{1/2}[/tex]

which is certainly true.
 
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  • #5
micromass said:
OK, but if ##A## is hermitian, then ##A^H = A##, no? So you can write your original post as

[tex]A = A^{1/2}A^{1/2}[/tex]

which is certainly true.

Oh, I didn't see it that way. Thanks
 

1. Can you explain what a Hermitian matrix is?

A Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose. This means that the entries on the main diagonal are real numbers, and the entries above and below the main diagonal are complex conjugates of each other.

2. What does the notation A=A^(1/2)AH^(1/2) mean?

This notation represents the decomposition of a Hermitian matrix A into the product of two matrices, A^(1/2) and AH^(1/2). The superscript 1/2 indicates the square root, and the AH represents the conjugate transpose of A.

3. Why is it important to be able to write a Hermitian matrix in this form?

Writing a Hermitian matrix in the form A=A^(1/2)AH^(1/2) allows for easier manipulation and calculation of the matrix. It also reveals important properties of the matrix, such as its eigenvalues and eigenvectors.

4. Is it always possible to write a Hermitian matrix in this form?

Yes, it is always possible to write a Hermitian matrix in the form A=A^(1/2)AH^(1/2). This is because every Hermitian matrix has a unique decomposition into its eigenvalues and eigenvectors, which can then be used to construct the matrices A^(1/2) and AH^(1/2).

5. How is this concept used in real-world applications?

The decomposition of Hermitian matrices is used in various fields such as quantum mechanics, signal processing, and statistics. It is also used in machine learning algorithms for dimensionality reduction and data analysis.

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