I am trying to relate eigenvalues with singular values. In particular,

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A are within the range of the singular values of AIn summary, the conversation discusses the relationship between eigenvalues and singular values. The statement being explored is that for any eigenvalue of matrix A, it is within the range of the singular values of A. Efforts have been made to use Schur decomposition and permutation to order the eigenvalues and singular values, but the relationship between them cannot be determined. It is noted that singular values exist for both square and rectangular matrices, while eigenvalues only exist for square matrices. It is also mentioned that the singular values of A are equal to the square roots of the eigenvalues of A^T.A. The conversation concludes by suggesting to prove that the eigenvalues of A are
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I am trying to relate eigenvalues with singular values. In particular, I'm trying to show that for any eigenvalue of A, it is within range of the singular values of A. In other words,

smallestSingularValue(A) <= |anyEigenValue(A)| <= largestSingularValue(A).

I've tried using Schur decomposition, and then permuting the matrix so that the eigenvalues are ordered like the singular values. But I can't determine their relationship. Any help would be appreciated.
 
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The singular values of A are equal to square roots of the eigenvalues of A^T.A.

Eigenvalues only exist for square matrices. Singular values exist for rectangular matrices as well as square ones.
 
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well, this is true:
smallestSingularvalue(T)*|v| <= |Tv| <= largestsingularvalue(T)*|v|
try proving that
 

Related to I am trying to relate eigenvalues with singular values. In particular,

1. What are eigenvalues and singular values?

Eigenvalues and singular values are mathematical concepts used in linear algebra to describe the properties of a matrix. Eigenvalues are the values that satisfy the equation Av = λv, where A is a square matrix, v is a non-zero vector, and λ is a scalar value. Singular values, on the other hand, are the square roots of the eigenvalues of the matrix A*A, where A* denotes the conjugate transpose of A.

2. How are eigenvalues and singular values related?

Eigenvalues and singular values are related through the singular value decomposition (SVD) of a matrix. SVD breaks down a matrix into three components: a unitary matrix, a diagonal matrix with singular values, and another unitary matrix. The singular values in the diagonal matrix are the square roots of the eigenvalues of the matrix A*A. This relationship is important in understanding the properties of a matrix and in various applications in data analysis and signal processing.

3. What is the significance of eigenvalues and singular values?

Eigenvalues and singular values provide important information about the properties of a matrix, such as its rank, invertibility, and stability. They also have applications in various fields, such as physics, engineering, and computer science. For example, in data analysis, eigenvalues and singular values are used to reduce the dimensionality of a dataset and to identify important features or patterns.

4. How do eigenvalues and singular values differ?

Eigenvalues and singular values differ in their definitions and properties. Eigenvalues are defined for square matrices, while singular values can be calculated for any matrix. Eigenvalues are complex numbers, while singular values are always positive real numbers. Additionally, the eigenvectors associated with eigenvalues are not necessarily orthogonal, while the singular vectors associated with singular values are always orthogonal.

5. Can you give an example of how eigenvalues and singular values are used in real-world applications?

One example of the use of eigenvalues and singular values is in image compression. In this application, a large matrix representing an image is decomposed into its singular values and vectors. The singular values are then sorted and the corresponding singular vectors are used to reconstruct the image with fewer dimensions. This reduces the storage space needed for the image without significantly affecting its quality.

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