Question 1161345
Let A= [−47 −90 ; 27 52]. 
Find S, D, and S^−1 such that A=SDS^−1
<pre>
We diagnalize the matrix:

{{{matrix(1,3,A,""="",
(matrix(2,2,-47,-90,27,52)))}}}

We find the eigenvalues

{{{abs(matrix(2,2,-47-lambda,-90,27,52-lambda)))}}}{{{""=""}}}{{{0}}}

{{{(-47-lambda)(52-lambda)-(-90)(27)=0}}}

{{{-2444-5lambda+lambda^2+2430=0}}}

{{{lambda^2-5lambda-14=0}}}

{{{(lambda-7)(lambda+2)=0}}}

 λ-7=0;  λ+2=0   
   λ=7;   λ=-2

by writing A as 

{{{matrix(1,3,A,""="",SDS^(-1))}}}

where D is the diagonal matrix with the two eigenvalues on the 
main diagonal:

{{{D = (matrix(2,2,7,0,0,-2))}}}

and the matrix S is 

{{{S=(matrix(1,2,V[1],V[2]))}}}

where the V's are the two column eigenvectors for the two eigenvalues

We find V<sub>1</sub> which is the eigengvector for the eigenvalue λ=7.

We find solutions for

{{{(A-7I)X=0}}}

{{{((matrix(2,2,-47^"",-90^"",27^"",52^""))^""^""-(matrix(2,2,7^"",0^"",0^"",7^"")))(matrix(2,1,x[1],x[2]))=(matrix(2,1,0^"",0^""))}}}

{{{(matrix(2,2,-54^"",-90^"",27^"",45^""))(matrix(2,1,x[1],x[2]))=(matrix(2,1,0^"",0^""))}}}

{{{-54x[1]-90x[2]=0}}}
Divide thru by -18
{{{3x[1]+5x[2]=0}}}
{{{3x[1]=-5x[2]}}}

We can take x<sub>1</sub>=1 and x<sub>1</sub>=1

So 

{{{v[1]=(matrix(2,1,-5,3))}}}

Now we do the same for the other eigenvalue

---

We find solutions for

{{{(A+2I)X=0}}}

{{{((matrix(2,2,-47^"",-90^"",27^"",52^""))^""^""-(matrix(2,2,-2^"",0^"",0^"",-2^"")))(matrix(2,1,x[1],x[2]))=(matrix(2,1,0^"",0^""))}}}

{{{(matrix(2,2,-45^"",-90^"",27^"",54^""))(matrix(2,1,x[1],x[2]))=(matrix(2,1,0^"",0^""))}}}

{{{-45x[1]-90x[2]=0}}}
Divide thru by -45
{{{x[1]+2x[2]=0}}}
{{{x[1]=-2x[2]}}}

We can take x<sub>1</sub>=1 and x<sub>2</sub>=1

So 

{{{v[2]=(matrix(2,1,-2,1))}}}

So

{{{S=(matrix(2,2,-5,3,-2,1))}}}

And since the determinant of S is 1, to find S<sup>-1</sup> we only
need to swap the elements on the the main diagonal and change the
signs of the other two elements"

{{{S^(-1)=(matrix(2,2,1,2,-3,-5))}}}

Then 

{{{A=SDS^(-1)}}}

{{{matrix(1,5,
A,
""="",
(matrix(2,2,-47^"",-90^"",27^"",52^"")),
""="",
(matrix(2,2,-5^"",-2^"",3^"",1^""))*(matrix(2,2,7,0^"",0^"",-2))*(matrix(2,2,1^"",2^"",-3^"",-5^""))

)}}}

Edwin</pre>