Singular Values & Linear Transformations

In summary, The conversation discusses the relationship between the singular values of matrices A and B, which are related by a non-singular linear transformation T. It is determined that the singular values of A and B are identical if T is orthogonal/unitary, and there may be a way to relate the singular values for the case where T is full-rank but not orthogonal/unitary. However, this relationship does not hold if T is not full-rank.
  • #1
grawil
3
0
I'm struggling to grasp what should be a trivial property of singular value decomposition. Say that I have a linear transformation T that is non-singular (i.e. [tex]T^{-1}[/tex] exists) and relates matrices A and B:

[tex]B = T A[/tex]

or

[tex]A = T^{-1} B[/tex]

What I would like to know is how the singular values of A and B are related?

A and B are both dimension (m x n), i.e. T is square (m x m). My thoughts are:

[tex]A = U_A \Lambda_A V_A^T[/tex]

[tex]B = U_B \Lambda_B V_B^T[/tex]

where U is (m x m), [tex]\Lambda[/tex] is (m x n) and V is (n x n). Substitution gives

[tex]T A = U_B \Lambda_B V_B^T[/tex]

[tex]T U_A \Lambda_A V_A^T = U_B \Lambda_B V_B^T[/tex]

but I'm not sure where to go next. From the above, can I simply write:

[tex]T U_A = U_B[/tex]

[tex]\Lambda_A = \Lambda_B[/tex]

[tex]V_A = V_B[/tex]

This implies the singular values are identical (with the caveat that T is non-singular) and that only the basis vectors of A are affected by the transformation... this is what I want to show but I'm unsure if I've actually succeeded with the above.
 
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  • #2
After a bit more thought... the above is only true if T is orthogonal/unitary. I still think it is possible to relate the singular values of A and TA for the slightly more general case where T is full-rank (i.e. non-singular) but I'm not entirely sure how. Obviously, if T is not of full rank then all bets are off.
 

1. What are singular values?

Singular values are a key concept in linear algebra and represent the square roots of the eigenvalues of a positive semidefinite matrix. They are used to determine the properties of a linear transformation, such as its scaling and rotational effects.

2. How are singular values calculated?

Singular values are calculated using the singular value decomposition (SVD) method. This involves breaking down a matrix into three components: a unitary matrix, a diagonal matrix, and the conjugate transpose of the unitary matrix. The diagonal elements of the diagonal matrix are the singular values.

3. What is the relationship between singular values and eigenvectors?

Singular values and eigenvectors are closely related. The singular values of a matrix are the square roots of the eigenvalues of the matrix's positive semidefinite part. Additionally, the left and right singular vectors of a matrix correspond to the left and right eigenvectors of its positive semidefinite part.

4. How do singular values affect a linear transformation?

Singular values play a crucial role in understanding the properties of a linear transformation. They determine the scaling and rotational effects of the transformation, and can be used to calculate the determinant, rank, and condition number of the matrix representing the transformation.

5. What are some applications of singular values and linear transformations?

Singular values and linear transformations have a wide range of applications in various fields, including data compression, image processing, machine learning, and quantum mechanics. They are also used in computer graphics, signal processing, and control systems.

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