Do exist different upper triangular matrices sharing the same singular values?

Question

Do exist different upper triangular matrices sharing the same singular values?

For example, I have the upper triangular matrix

$$\begin{pmatrix} 10 & 5 & 30\\ 0 & 3 & 10\\ 0 & 0 & 15 \end{pmatrix}$$

and with Singular Value Decomposition, I find that it has a singular vector $$(36.427657, 5.118751, 2.413333)$$

So, my concern is, are there any other upper triangular matrices

$$\begin{pmatrix} a & b & c\\ 0 & d& e\\ 0 & 0 & f \end{pmatrix}$$

with the same singular values?

Answer 1

We can easily find other upper triangular matrices with the same singular values as follows.

Let indicate with $A$ the upper triangular matrix in hand and $M$ is an upper triangular orthogonal matrix (i.e. diagonal), then $MA$ is upper triangular too, moreover $MAA^TM^T=MAA^TM^{-1}$ is similar to $AA^T$ and then $MA$ share with $A$ the same singular values.

For example:

$$\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 10 & 5 & 30\\ 0 & 3 & 10\\ 0 & 0 & 15 \end{pmatrix}=\begin{pmatrix} 10 & 5 & 30\\ 0 & -3 & -10\\ 0 & 0 & -15 \end{pmatrix}$$

For the general form of triangular orthogonal matrices, refer to:

• Show which matrices are upper triangular orthogonal in $\mathbb R$.