# Singular values of unitarily equivalent matrices

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In summary, If two square matrices, A and B are unitarily equivalent, then A = QBQ* for some unitary Q of the same size as A and B. To prove that A and B are unitarily equivalent, we start from the definition of singular values and substitute A with QBQ*. We then operate on the left by Q* and use the fact that Q is unitary (Q*=Q^(-1)). This leads to showing that the singular values of A and B are the same.
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## Homework Statement

If two square matrices, A and B are unitarily equivalent then A = QBQ* for some unitary Q of the same size as A and B. Prove that A and B are unitarily equivalent if and only if they have the same singular values

## The Attempt at a Solution

I started from the definition of singular values:

Au = sigma v for singular vectors u and v
A*v = sigma u
substitution A with QBQ*,
QBQ*u = sigma v
(QBQ*)*v = QB*Q*v = sigma u

Can't see how this leads to proving that the sigmas of A and B are the same..?

If Q is unitary, the Q*=Q^(-1). Operate on the left by Q*.

## 1. What are singular values of unitarily equivalent matrices?

Singular values of unitarily equivalent matrices are the same for two matrices that are related by a unitary transformation. A unitary transformation is a linear transformation that preserves the length of vectors and the angle between them. Essentially, this means that the singular values of unitarily equivalent matrices are the same because the matrices have the same "shape", just represented in a different coordinate system.

## 2. How can I calculate the singular values of unitarily equivalent matrices?

The singular values of unitarily equivalent matrices can be calculated by first finding the eigenvalues of the matrices, taking the absolute value of each eigenvalue, and then sorting them in descending order. The resulting values are the singular values of the matrices.

## 3. What is the significance of the singular values of unitarily equivalent matrices?

The singular values of unitarily equivalent matrices are important in understanding the geometry and properties of the matrices. They can provide information about the scaling and rotation of the matrices, and can also be used to determine the condition number of a matrix, which measures its sensitivity to perturbations.

## 4. Can singular values of unitarily equivalent matrices be negative?

No, singular values cannot be negative. They are always non-negative real numbers. However, they can be equal to zero, which indicates that the matrix is singular and cannot be inverted.

## 5. How are singular values of unitarily equivalent matrices related to the singular values of the original matrices?

The singular values of unitarily equivalent matrices are the same as the singular values of the original matrices. This is because unitary transformations do not change the eigenvalues or the absolute values of the eigenvalues, only the coordinate system in which they are represented.

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