How many SVD's does a matrix have?

Question

If $A$ is a $3 \times 3$ matrix with singular values $5$, $4$, and $2$, then there are $9$ distinct singular value decompositions of $A$. True or false?

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Is there any method to solve this because I'm not sure how to approach this.

Answer 1

Important rule:

• An $n*n$ matrix with n distinct positive singular values has $2^{n}$ different singular value decompositions (svds).

• An $n*n$ singular matrix (one without an inverse) with $n$ distinct singular values ($\sigma_{n}$ can be $0$) has $2^{n+1}$ distinct svds.

• An $n*n$ matrix with a repeated singular value has $\infty$ svds.

This problem satisfies the first rule, therefore $A$ has $2^{3}=8$ distinct svds.

Answer: False