Can orthonormal columns lead to orthonormal rows in a square matrix?

In summary, a unitary matrix is a square matrix that has the property of being equal to its conjugate transpose and inverse. It is used in linear algebra and quantum mechanics to represent transformations, and also has applications in coding theory and signal processing. It differs from an orthogonal matrix in that it is defined over complex numbers and can have complex entries. To determine if a matrix is unitary, you can multiply it by its conjugate transpose and see if the result is the identity matrix. Unitary matrices also have real-world applications in fields such as computer graphics, image processing, and machine learning.
  • #1
td21
Gold Member
177
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"orthonormal columns imply orthonormal rows for square matrix."

My proof is:
[itex]Q^{T}Q=I[/itex](orthonormal columns)
implys
[itex]QQ^{T}=I[/itex](orthonoraml rows)

for square matrix.


But i think this proof is kind of indirect. Is there another more direct proof from the definition of inner product or norm?
 
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  • #2
I think that's as direct and simple as it gets. I would only add the explanation of how [itex]Q^TQ=I[/itex] implies [itex]QQ^T=I[/itex].
 

1. What is a unitary matrix?

A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. This means that multiplying a unitary matrix by its conjugate transpose will result in the identity matrix.

2. What is the purpose of a unitary matrix?

Unitary matrices are used in linear algebra and quantum mechanics to represent transformations that preserve the magnitude and angles between vectors. They are also used in coding theory and signal processing.

3. How is a unitary matrix different from an orthogonal matrix?

While both unitary and orthogonal matrices have the property of preserving the magnitude and angles between vectors, unitary matrices are defined over complex numbers while orthogonal matrices are defined over real numbers. Additionally, unitary matrices may have complex entries while orthogonal matrices can only have real entries.

4. How do you determine if a matrix is unitary?

To determine if a matrix is unitary, you can compute its conjugate transpose and multiply it by the original matrix. If the result is the identity matrix, then the original matrix is unitary.

5. What are some real-world applications of unitary matrices?

Unitary matrices are used in various fields such as quantum mechanics, signal processing, and coding theory. They are also used in computer graphics and image processing to rotate and reflect objects. Additionally, they are used in machine learning algorithms for dimensionality reduction and data compression.

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