# Eigenvalues of $A$ based on $a$


```julia
using LinearAlgebra
function A(a)
    eigvals([1 a; a 1])
end

A(1)
```




    2-element Array{Float64,1}:
     0.0
     2.0



$
\begin{align}
det(A-\lambda I) &= 0 \\
(1-\lambda)^2 - a^2 &= 0 \\
\lambda^2 - 2 \lambda +  (1- a^2) &=0 \\
\rightarrow \lambda &= \frac{1 \pm \sqrt{1 - 1 +a^2}}{1} \\
&= 1 \pm \| a\|
\end{align}
$

# SVD and Eigenvalue

$k = \frac{\sigma_{max}}{\sigma_{min}}$

$a=1$ is making $A$ a singular matrix. 

# Quadratic form


```julia
using Plots

function quadratic(u)
    range = collect(0:0.01:2)
    outpt = zeros(size(range,1))
    for n in 1:size(range,1)
        a = range[n]
        y =  u' * [1 a; a 1] * u
        outpt[n] = y
    end
    plot(range, outpt, xlabel="a", ylabel="f(u)"
        , title="u="*string(u), label=:false, linewidth=4)
end

quadratic([5, 2])
```




    

    



For $a=1$, the output of $f(u)$ function is $(u[1]+u[2])^2$;

## Adjourn


```julia
using Dates
println("mahdiar")
Dates.format(now(), "Y/U/d HH:MM")  
```

    mahdiar





    "2021/February/11 17:54"



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ipynb
Jupyter Notebook
HW03/2.ipynb
mahdiarsadeghi/NumericalAnalysis
95a0914c06963b0510971388f006a6b2fc0c4ef9
MIT
HW03/2.ipynb
mahdiarsadeghi/NumericalAnalysis
95a0914c06963b0510971388f006a6b2fc0c4ef9
MIT
HW03/2.ipynb
mahdiarsadeghi/NumericalAnalysis
95a0914c06963b0510971388f006a6b2fc0c4ef9
MIT
Qwen/Qwen-72B
1. YES
2. YES