求解矩阵方程A=BCD中的C：含正交与对角矩阵场景

Nice question! Let's break this down both in the general case and the specific scenario you're working with—those orthogonal/diagonal constraints are way more meaningful than you might initially think.

1. General Case (No Orthogonal/Diagonal Constraints) #

Without any special properties for ( B ), ( C ), or ( D ), solving for ( C ) hinges on the invertibility of ( B ) and ( D ):

• If ( B ) is square and full-rank (invertible) and ( D ) is square and full-rank (invertible), we can rearrange the equation directly:
  C = B⁻¹ A D⁻¹
• If ( B ) or ( D ) are non-square (but still dimensionally compatible with the multiplication), we need to use the Moore-Penrose pseudoinverse (⁺) instead:
  C = B⁺ A D⁺
  Note: In this non-square scenario, there may be infinitely many valid solutions for ( C ) unless additional constraints are imposed.

2. Special Case (B & D Orthogonal, C Diagonal) #

This is your actual application scenario, and these constraints are not optional—they make the solution concrete and tie it to a fundamental matrix decomposition:

First, remember a key property of orthogonal matrices: their inverse equals their transpose, so ( B⁻¹ = Bᵀ ) and ( D⁻¹ = Dᵀ ). Rearranging the original equation gives:
C = Bᵀ A Dᵀ

Since ( C ) must be diagonal, this means ( Bᵀ A Dᵀ ) has to be a diagonal matrix. This is exactly what the Singular Value Decomposition (SVD) delivers for any matrix ( A ):

For any matrix ( A ), there exist orthogonal matrices ( U ) and ( V ) such that ( A = U Σ Vᵀ ), where ( Σ ) is a diagonal matrix containing the singular values of ( A ) (sorted in non-increasing order).

Mapping this to your equation:

• ( B ) corresponds to the orthogonal matrix ( U )
• ( D ) corresponds to ( Vᵀ ) (which is also orthogonal, since ( V ) is orthogonal)
• ( C ) corresponds to the diagonal singular value matrix ( Σ )

Here's how to apply this in practice:

• If you need to find both the orthogonal matrices ( B/D ) and the diagonal ( C ): Perform an SVD on ( A ). The ( Σ ) from the decomposition is your ( C ), ( U ) is ( B ), and ( Vᵀ ) is ( D ).
• If ( B ) and ( D ) are already given (fixed orthogonal matrices): Compute C = Bᵀ A Dᵀ and check if it's diagonal. If yes, that's your solution; if not, no diagonal matrix ( C ) exists that satisfies ( A = BCD ) with those specific ( B ) and ( D ).

This special case is incredibly useful in applications like dimensionality reduction, image compression, or signal processing, where orthogonal transformations preserve distances and diagonal matrices simplify calculations drastically.

内容的提问来源于stack exchange，提问作者Jack Elsey