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Digital Juice .Q:

Relationship between eigenvalues and singular values

I have a question related to the relationship between singular values and the eigenvalues of a symmetric matrix.
A(a,b,a,b,0,..,..,0,0) is a symmetric matrix that has eigenvalues $\pm\sqrt{4ab}$. In the text they say that it is clear that the singular values will be $\sqrt{2ab}$ and $\sqrt{2ab}$ because of the matrix. Could somebody please explain this to me? I do not understand why we say that?
Thanks in advance for the help.

A:

This is because the matrix is diagonalized by two orthogonal matrices. In the symmetric form this means that the symmetric matrix is diagonalized by an orthogonal matrix and its positive semidefinite square root:
$$
\pmatrix{4ab&0&0&\cdots&0&0\\0&0&0&\cdots&0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots&0&0\\0&0&0&\cdots&0&0}=\pmatrix{s_{\uparrow}&0&0&\cdots&0&0\\0&s_{\downarrow}&0&\cdots&0&0\\\vdots&\vd

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before they are written out to permanent storage. The important thing is to verify that all the constraint methods have been called. Any errors that occur during the verification phase can’t be overlooked or treated as “unimportant”.

Using the Verification Phase to Check Constraints

Once again, the built in constraint checking techniques are available to us. There are two things we can do in the verification phase:

Assert that all the relevant constraints were met (Called “Equals”)

Check that the structure of the constraint is satisfied (Called “Subset”)

The Subset check will not be performed unless it is enabled for a specific constraint. It can be enabled for an individual constraint by passing an extra parameter to the constraint constructor (e.g. PatternProperties.Subset(new KeyValuePair(“Age”,22))).

In the case above, we would have to provide a value for the key “Age”. This value would then be used to test the constraint when the method is called.

So how do we deal with missing or “null” values?

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FluentAssertions includes special handling for null values. If a value is missing, any constraint method like HasItem() will return null instead of an empty collection. Any Arrange() or Verify() calls that depend on those collections will also return null. This means that the test code has to be aware of these possibilities and code for
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