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Edit 4

To capture what I said in some additional comments: if you do decide to use the lowest singular value as a merit function for further root finding or minimization, as a function of the parameter κ, then you may run into the problem that SingularValueList will omit the lowest singular values if they are too small relative to the largest singular value. This can lead to jumps if you plot the lowest singular value versus κ, because you aren't tracking the same singular value at all times. To insure that no singular values are omitted, one has to use

SingularValueList[m, Tolerance -> 0]

I can't answer the question of whether this is physically the best procedure, because I don't know what the physical problem is.

A very good mathematical discussion of the singular value decomposition is in Numerical Recipes in C on page 59.

The pattern f[κ_?NumericQ] insures that the function will return a value only if it is called with a numerical argument κ. Any minimization or root finding routine will by default attempt to simplify f[κ] symbolically, but since f is defined_only_ for numerical arguments that attempt will fail. This is good because the automatic evaluator then chooses a purely numerical method, which in our case is faster.

So there's still some work to do to get rid of the warning message, but in principle we got somewhere.

The pattern f[κ_?NumericQ] insures that the function will return a value only if it is called with a numerical argument κ. Any minimization or root finding routine will by default attempt to simplify f[κ] symbolically, but since f is defined only for numerical arguments that attempt will fail. This is good because the automatic evaluator then chooses a purely numerical method, which in our case is faster.

So there's still some work to do to get rid of the warning message, but in principle we got somewhere.

Edit 3

In order to make a pre-existing matrix mat (containing a parameter κ) into a function that you can use for root finding etc., you have to watch out for the difference between the global variable κ and the pattern instance κ that is passed to the function in place of the "dummy variable" in the pattern f[κ_?NumericQ].

The dummy variable could have equally well been named x_ or b_, so how do we make sure that it gets substituted into the matrix wherever the global variable κ appears?

Here is a way to do that. Assume the matrix is externally given as

Clear[κ]; mat = {{-892.33`, 973.21`, 44.306` + κ, -81.103`, 
    0}, {446.12`, -557.94`, 0, -682.54`, -314.89`}, {0, 
    893.37`, -506.68` κ, -391.457`, 0}, {0, 429.78`, 
    0, -210.47`, 342.85`}, {278.32` κ, 0, 963.41`, 
    217.71`, -342.68` + κ}};

Then change your function definition to

f[x_?NumericQ] := Last[SingularValueList[mat /. κ -> x]]

What this does is to replace the global κ (which you must make sure is unassigned, hence the Clear for safety) by the value of the dummy variable in the function. I have to change its name to something other than κ, so I chose x.

Last@DSingularValueList[m]

Last@SingularValueList[m]