# Singular Values and Eigenvalues

• MHB
• linearishard
In summary, the conversation discusses the relationship between a matrix's eigenvalues and singular values, specifically in symmetric positive definite matrices. The Spectral Theorem is mentioned, stating that every real symmetric matrix is diagonalizable. The concept of positive definiteness is also brought up, and it is noted that while the forward conditional is true (symmetric positive definite matrix has positive eigenvalues), the opposite is not necessarily true.
linearishard
Hi, one more question!

How do I prove that A has eigenvalues equal to its singular values iff it is symmetric positive definite? I think I have the positive definite down but I can't figure out the symmetric part. Thanks!

It will help if you post the work you already have.

All I have is that if a singular value is the eigenvalue of ATA, then A must be positive semi definite or the signs will be different on at least one eigenvalue. I don't know where to start with symmetry or if my assumption is correct.

Are you familiar with the Spectral Theorem?
That is that every real symmetric matrix is diagonalizable?

So if the matrix is symmetric and has real numbers as its elements, it is diagonalizable, which means that it has a full set of real eigenvalues and corresponding eigenvectors that span the vector space.

Hello again, linearishard,

Usually, positive definite matrices are assumed to be symmetric (or Hermitian for complex matrices). Does your teacher's definition of positive definite exclude the symmetry assumption? In any case, the problem statement is not true. For since singular values of a matrix can be zero, having eigenvalues of $A$ equal to the singular values of $A$ does not necessarily result in every eigenvalue being positive (which is what you need to claim positive definiteness).

The forward conditional is true, however. if $A$ is symmetric positive definite, the eigenvalues of $A$ are positive. The eigenvalues of $A$ are square roots of the eigenvalues of $A^2 = A^TA$, so the singular values of $A$ are the eigenvalues of $A$.

## 1. What is the difference between singular values and eigenvalues?

Singular values and eigenvalues are both mathematical concepts related to matrices. The main difference between them is that eigenvalues are associated with square matrices, while singular values are associated with any type of matrix.

## 2. How do singular values and eigenvalues relate to each other?

Singular values and eigenvalues are related through the singular value decomposition (SVD) of a matrix. In SVD, a matrix is decomposed into three matrices, one of which contains the eigenvalues of the original matrix. This means that the eigenvalues of a matrix are also its singular values.

## 3. What are the applications of singular values and eigenvalues?

Singular values and eigenvalues have many applications in various fields such as data science, image processing, and engineering. They are used in dimensionality reduction techniques, signal processing, and determining the stability of a system, among others.

## 4. How are singular values and eigenvalues calculated?

The calculation of singular values and eigenvalues involves complex mathematical operations and algorithms. Singular values can be calculated using the SVD method, while eigenvalues can be found by solving the characteristic equation of a square matrix.

## 5. What is the significance of singular values and eigenvalues in linear algebra?

Singular values and eigenvalues play a crucial role in linear algebra as they provide information about the properties of a matrix. They help in understanding the behavior of a matrix and can be used for various operations such as matrix inversion and diagonalization.

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