Suppose there exist matrices $A \in \Bbb C^{m \times n}$ and $B = U_1 A U_2$, where $U_1 \in \Bbb C^{m \times m}$ and $U_2 \in \Bbb C^{n \times n}$ are unitary matrices (not necessarily related to each other). Then, $A$ and $B$ have the same singular values.

The reason for this is because $A^H A$ and $B^H B$ are related the following way ($A^H$ is the Hermitian transpose of $A$): $$B^H B = (U_1 A U_2)^H (U_1 A U_2) = U_2^H A^H U_1^H U_1 A U_2 = U_2^H A^H A U_2 = U_2^{-1} (A^H A) U_2$$ Since $A^H A$ and $B^H B$ are similar matrices (by definition), they share the same eigenvalues. Since the singular values of any matrix $M$ are the positive square-roots of the eigenvalues of $M^T M$, $A$ and $B$ have the same singular values.

Is there a special name relating these types of matrices (just like "similar matrix" relates $A$ and $B$ with the same eigenvalues)? I know that orthogonally equivalent matrices $C$ and $D = U C U^H$ are kind of similar to this, but $C$ and $D$ are always square, and the two unitary matrices are related as the inverse of one another ($U^H = U^{-1}$). In my problem statement, $A$ and $B$ can be rectangular, and the 2 unitary matrices $U_1$ and $U_2$ don't have to be related to each other (in fact, even their dimensions may differ).


3 Answers 3


If $B=UAV$ for some unitary matrices $U$ and $V$ (of possibly different sizes), $A$ and $B$ are said to be unitarily equivalent. Note that unitary equivalence is really an equivalence relation.

However, beware that quite a lot of authors also use the term "unitarily equivalent" to mean "unitarily similar".

  • $\begingroup$ Doesn't Unitary Equivalence also state that the two unitary matrices are inverses of one another? <en.wikipedia.org/wiki/…*AU.&text=Two%20similar%20matrices%20represent%20the,basis%20to%20another%20orthonormal%20basis.> $\endgroup$
    – 5Pack
    Aug 8, 2020 at 0:54
  • 1
    $\begingroup$ @5Pack Please see the second paragraph of my answer. $\endgroup$
    – user1551
    Aug 8, 2020 at 0:58
  • $\begingroup$ could you reference an instance in literature where "unitarily equivalent" is defined with 2 unrelated unitary matrices $U$ and $V$? $\endgroup$
    – 5Pack
    Aug 8, 2020 at 1:03
  • 3
    $\begingroup$ @5Pack See for instances Gilbert W. Stewart's Introduction to Matrix Computations, Lancaster and Tismenetsky's The Theory of Matrices with Applications, Kenneth H. Rosen's (ed.) Handbook of Linear Algebra, or Helmut Lütkepohl's Handbook of Matrices. $\endgroup$
    – user1551
    Aug 8, 2020 at 1:26

On page 381 of the solutions manual to "Introduction to Linear Algebra" by Gilbert Strang ($5^{th}$ edition), Strang states,

"The matrices $A$, $Q_1 A Q_2^T$, and $\Sigma$ are all 'isometric' $=$ sharing the same $\Sigma$.

($Q_1$ and $Q_2$ = unitary, $\Sigma$ = diagonal matrix of singular values)


Not really an answer but a suggestion: Your question is obviously very much related to the Singular value decomposition In the SVD a complex matrix $A$ is written as $A = U M V^*$ with $U$ and $V$ unitary. As your definition is equivalent to saying that the two matrices $A$ and $B$ admit SVD with the same $M$ you could say that your two matrices are SVD-equivalent?


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