Find the S + N decompositon of these three matrices.  You may use only the Eigenvalues[ ], Inverse[ ] and RowReduce[ ] commands, as well as MatrixForm[] to display your matrices.   Send me a Mathematica notebook or a Matlab file as an attachment.  There is no need to print it out.

A = {{-2,-1,-3},{-12,0,-8},{10,2,9}};
Print["A = ", MatrixForm[A]];

A = {{6,2,4},{10,7,10},{-8,-4,-6}};
Print["A = ", MatrixForm[A]];

A = {{6,5,-1,0},{-2,-1,0,0},{9,5,2,-2},{14,15,-2,1}}
Print["A = ", MatrixForm[A]];

Note:  The final example has repeated complex eigenvalues.  Do this problem twice, using both Theorem 1 and Theorem 2 of section 1.7 in the book.  For 4x4 real matrices with eigenvalues a + ib, a+ib, a-ib, a-ib, let P = [(Subscript[v, 1])  (Subscript[v, 2])  (Subscript[v, 3])   (Subscript[v, 4]) ]  , where where {(Subscript[v, j])} are linearly independent (over C) generalized eigenvectors in (C^2).  Let v1 and v2 be the generalized eigenvectors of a + i b.  Then, the simplest choice is to let v3 = v1-bar and v4 = v2-bar.  Use capital I for the complex unit.

I^2

Here is a sample of what I want you to do:

A = {{3,-1,1,0},{1,0,3,0},{5,-1,3,2},{0,2,-2,3}};
Print["A = ", MatrixForm[A]];

Eigenvalues[A]

A3 = A - 3IdentityMatrix[4]

MatrixForm@RowReduce[A3]

v1 = {0, 2,2,1};
A3.v1

A2 = A - 2IdentityMatrix[4]

MatrixForm@RowReduce[A2.A2.A2]

v2 = {0, 1, 0, 0};
v3 = {0, 0, 1, 0};
v4 = {-1, 0, 0,2};

Check that all these are 0.

A2.A2.A2.v2
A2.A2.A2.v3
A2.A2.A2.v4

P = Transpose[{v1, v2, v3, v4}];

S = P.DiagonalMatrix[{3,2,2,2}].Inverse[P];
MatrixForm[S]
Nil = A - S;
MatrixForm[Nil]

Nil.Nil
Nil.Nil.Nil

S.Nil - Nil.S