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I want to diagonalize a matrix by orthogonalization, but the following method cannot get the correct result:

A = {{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}};
DiagonalizableMatrixQ[A]
Eigensystem[A]
Q = Orthogonalize@(Eigensystem[A][[2]])

Transpose[Q].DiagonalMatrix[{6, 2, 2}].Q == A
Transpose[Q].A.Q == DiagonalMatrix[{6, 2, 2}]

I think it is probably caused by the difference between function Orthogonalize (Q = Orthogonalize@(Eigensystem[A][[2]])) and Schmidt orthogonalization.

How can I use built-in functions to do Schmidt orthogonalization and diagonalize the matrix?

Oddly enough, the following code returns an empty set, whereas DiagonalizableMatrixQ[A]==True indicates that the matrix can be diagonalized:

Q = Array[x, {3, 3}];
A = {{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}};
FindInstance[
 Q\[Transpose].A.Q == DiagonalMatrix[{6, 2, 2}] && 
  Q\[Transpose].Q == IdentityMatrix[3], Flatten[Q]]

I hope to find the orthogonal matrix Q by Schmidt orthogonalization and diagonalize the matrix A.

The definition of Schmidt orthogonalization:

$$\begin{array}{l} \boldsymbol{b}_{1}=\boldsymbol{a}_{1} \\ \boldsymbol{b}_{2}=\boldsymbol{a}_{2}-\frac{\left[\boldsymbol{b}_{1}, \boldsymbol{a}_{2}\right]}{\left[\boldsymbol{b}_{1}, \boldsymbol{b}_{1}\right]} \boldsymbol{b}_{1} \\ \cdots \cdots \cdots \cdots \\ \boldsymbol{b}_{r}=\boldsymbol{a}_{r}-\frac{\left[\boldsymbol{b}_{1}, \boldsymbol{a}_{r}\right]}{\left[\boldsymbol{b}_{1}, \boldsymbol{b}_{1}\right]} \boldsymbol{b}_{1}-\frac{\left[\boldsymbol{b}_{2}, \boldsymbol{a}_{r}\right]}{\left[\boldsymbol{b}_{2}, \boldsymbol{b}_{2}\right]} \boldsymbol{b}_{2}-\cdots-\frac{\left[\boldsymbol{b}_{r-1}, \boldsymbol{a}_{r}\right]}{\left[\boldsymbol{b}_{r-1}, \boldsymbol{b}_{r-1}\right]} \boldsymbol{b}_{r-1}, \end{array}$$

$$\boldsymbol{e}_{1}=\frac{1}{\left\|\boldsymbol{b}_{1}\right\|} \boldsymbol{b}_{1}, \quad \boldsymbol{e}_{2}=\frac{1}{\left\|\boldsymbol{b}_{2}\right\|} \boldsymbol{b}_{2}, \quad \cdots, \quad \boldsymbol{e}_{r}=\frac{1}{\left\|\boldsymbol{b}_{r}\right\|} \boldsymbol{b}_{r}$$

$[x,y]$ is the inner product of vectors x and y.

Other examples for testing:

A = {{-2, 1, 1}, {0, 2, 0}, {-4, 1, 3}};
DiagonalizableMatrixQ[A]
Q = (Eigenvectors[A])
Inverse[Q].A.Q
A.Q

I don't know why finding eigenvector matrix does not satisfy $\boldsymbol{A} \boldsymbol{p}_{i}=\lambda_{i} \boldsymbol{p}_{i} \quad(i=1,2, \cdots, n)$ or $Q^{-1} A Q=\Lambda$.

The eigenvector matrix $\boldsymbol{Q}=\left(\boldsymbol{p}_{1}, \boldsymbol{p}_{2}, \boldsymbol{p}_{3}\right)=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 4 \end{array}\right)$ obtained in the textbook can satisfy $\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 4 \end{array}\right)^{-1} \cdot A \cdot\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 4 \end{array}\right)=\left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right)$.

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    $\begingroup$ The result of SchurDecomposition[N[{{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}}]] tells me that your desire to have orthonormal eigenvectors is not reasonable. $\endgroup$ Aug 19, 2020 at 9:06
  • 2
    $\begingroup$ Schmidt orthogonalization is done in Mathematica with the function QRDecomposition. $\endgroup$ Aug 19, 2020 at 9:33
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    $\begingroup$ Orthogonalize by default generates a Gram[Dash]Schmidt basis. $\endgroup$ Aug 19, 2020 at 10:12
  • $\begingroup$ @J.M. Thank you very much. I've learned that DiagonalizableMatrixQ[A]==True indicates that A can be diagonalized, but SchurDecomposition[N[{{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}}]] means that A cannot be orthogonal diagonalized. $\endgroup$ Aug 20, 2020 at 0:48

1 Answer 1

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It should be noted that these matrices are not symmetric matrices. In addition, eigenvector matrices need to be transposed before been used:

A = {{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}};
DiagonalizableMatrixQ[A]
Eigensystem[A]
Q = (Eigensystem[A][[2]])
(*but this matrix is not symmetric,so Transpose is not sufficient*)
Inverse[Q\[Transpose]].A.Q\[Transpose]


A = {{-2, 1, 1}, {0, 2, 0}, {-4, 1, 3}};
DiagonalizableMatrixQ[A]
Q = (Eigenvectors[A])
Inverse[Q\[Transpose]].A.Q\[Transpose]

GramSchmidtOrthogonalize[mat_] := 
   Module[{matT = Transpose[mat], 
       n = Length[Transpose[mat]]}, 
     a = matT; b[1] = a[[1]]; b[i_] := 
        a[[i]] - Sum[(b[j] . a[[i]]/b[j] . b[j])*b[j], 
            {j, 1, i - 1}]; Transpose[Table[b[i], 
          {i, 1, n}]]]
GramSchmidtOrthogonalize[{{2, 0, 0}, {1, 2, 0}, 
     {0, 0, -1}}]

I've learned that DiagonalizableMatrixQ[A]==True indicates that A can be diagonalized, but SchurDecomposition[N[{{1, 2, -3}, {-1, 4, -3}, {1, -2, 5}}]] means that A cannot be orthogonal diagonalized:

A = {{1, 2, -3}, {-1, 4, -3}, {1, a, 5}} /. a -> -2 ;
Q = Join[{Eigensystem[A][[2]][[1]]}\[Transpose], 
  GramSchmidtOrthogonalize[(Eigensystem[A][[2]][[2 ;; 
        All]])\[Transpose]], 2]
Inverse[Q].A.Q

GramSchmidtOrthogonalize[mat_] := 
 Module[{matT = Transpose[mat], n = Length[Transpose[mat]], a, b}, 
     a = matT; b[1] = a[[1]]; 
  b[i_] := a[[i]] - 
    Sum[(b[j] . a[[i]]/b[j] . b[j])*b[j], {j, 1, i - 1}]; 
      Transpose[Table[b[i], {i, 1, n}]]]
A = {{1, 2, -3}, {-1, 4, -3}, {1, a, 5}} /. a -> -2; 
Q = GramSchmidtOrthogonalize[Eigensystem[A][[2]]]; Normalize /@ 
 Transpose[Q]
Orthogonalize[Transpose[Eigenvectors[A]]]
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