A 3X3 example

mA = {{-2, 2, 1}, {2, 1, 2}, {1, 2, -2}}

Eigensystem[mA]

{{-1, 0, 1}, {-2, 1, 0}, {1, 2, 
        1}}

But theses vectors are not orthogonal.

U1 = (1/(Sqrt[2])) {{-1}, {0}, {1}};
U2 = (1/(Sqrt[3])) {{1}, {-1}, {1}};

    
U3 = (1/(Sqrt[6])) {{1}, {2}, {1}}

(-3) U1 . Transpose[U1] + (-3) U2 . Transpose[U2] + 
      3 U3 . Transpose[U3]

Exactly the matrix mA, or more convincing.

(-3) U1 . Transpose[U1] + (-3) U2 . Transpose[U2] + 
      3 U3 . Transpose[U3] - mA

4X4 examples

Example 1

mA1 = {{1, -1, 1, 1}, {-1, 1, 1, 1}, {1, 1, -1, 1}, {1, 
            1, 1, -1}};

MatrixForm[mA1]

Eigensystem[mA1]

These are not particularly nice eigenvectors. Notice that the nicer eigenvectors are

mA1 . {1, 1, -1, -1}

mA1 . {0, 0, 1, -1}

mA1 . {1, 1, 1, 1}

mA1 . {1, -1, 0, 0}

Normalized and put in the matrix P

mP1 = 
      Transpose[{(1/2) {1, 1, -1, -1}, (1/(Sqrt[2])) {0, 0, 
              1, -1}, (1/2) {1, 1, 1, 1}, (1/(Sqrt[2])) {1, -1, 0, 
              0}}]

Transpose[mP1] . mA1 . mP1

U1 = Transpose[{Transpose[mP1][[1]]}]; 
    MatrixForm[U1]

U2 = Transpose[{Transpose[mP1][[2]]}]; 
    MatrixForm[U2]

U3 = Transpose[{Transpose[mP1][[3]]}]; 
    MatrixForm[U3]

U4 = Transpose[{Transpose[mP1][[4]]}]; 
    MatrixForm[U4]

(-2) U1 . Transpose[U1] + (-2) 
        U2 . Transpose[U2] + (2) U3 . Transpose[U3] + (2) 
        U4 . Transpose[U4]

Exactly the matrix mA1, or more convincing.

(-2) U1 . Transpose[U1] + (-2) 
        U2 . Transpose[U2] + (2) U3 . Transpose[U3] + (2) 
        U4 . Transpose[U4] - mA1

Example 2

mA2 = {{1, -1, 1, 1}, {-1, 1, 1, 1}, {1, 1, 1, -1}, {1, 
            1, -1, 1}}; 
MatrixForm[mA2]

Eigensystem[mA2]

These are not particularly nice eigenvectors. Notice that the nicer eigenvectors are

mA2 . {1, 1, -1, -1}

mA2 . {0, 0, 1, -1}

mA2 . {1, 1, 1, 1}

mA2 . {1, -1, 0, 0}

Normalized and put in the matrix P

mP2 = 
      Transpose[{(1/2) {1, 1, -1, -1}, (1/(Sqrt[2])) {0, 0, 
              1, -1}, (1/2) {1, 1, 1, 1}, (1/(Sqrt[2])) {1, -1, 0, 
              0}}]

Transpose[mP2] . mA2 . mP2

U1 = Transpose[{Transpose[mP2][[1]]}]; 
    MatrixForm[U1]

U2 = Transpose[{Transpose[mP2][[2]]}]; 
    MatrixForm[U2]

U3 = Transpose[{Transpose[mP2][[3]]}]; 
    MatrixForm[U3]

U4 = Transpose[{Transpose[mP2][[4]]}]; 
    MatrixForm[U4]

(-2) U1 . Transpose[U1] + (-2) 
        U2 . Transpose[U2] + (2) U3 . Transpose[U3] + (2) 
        U4 . Transpose[U4]

Exactly the matrix mA2, or more convincing.

(-2) U1 . Transpose[U1] + (2) U2 . Transpose[U2] + (2) 
        U3 . Transpose[U3] + (2) U4 . Transpose[U4] - mA2

Example 3

mA = {{-2, 2, 1}, {2, 1, 2}, {1, 2, -2}}

MatrixForm[mA]

mA3 = {{-1, 2, 2, 1}, {2, -1, 1, 2}, {2, 1, -1, 2}, {1, 
            2, 2, -1}}; 
MatrixForm[mA3]

Eigensystem[mA3]

These are particularly nice eigenvectors.

Normalized and put in the matrix P

mP3 = 
      Transpose[{(1/2) {1, -1, -1, 1}, (1/2) {1, 1, 1, 
              1}, (1/(Sqrt[2])) {-1, 0, 0, 1}, (1/(Sqrt[2])) {0, -1, 1, 
              0}}]

Transpose[mP3] . mA3 . mP3

U1 = Transpose[{Transpose[mP3][[1]]}]; 
    MatrixForm[U1]

U2 = Transpose[{Transpose[mP3][[2]]}]; 
    MatrixForm[U2]

U3 = Transpose[{Transpose[mP3][[3]]}]; 
    MatrixForm[U3]

U4 = Transpose[{Transpose[mP3][[4]]}]; 
    MatrixForm[U4]

(-4) U1 . Transpose[U1] + (4) 
        U2 . Transpose[U2] + (-2) U3 . Transpose[U3] + (-2) 
        U4 . Transpose[U4]

Exactly the matrix mA3, or more convincing.

(-4) U1 . Transpose[U1] + (4) 
        U2 . Transpose[U2] + (-2) U3 . Transpose[U3] + (-2) 
        U4 . Transpose[U4] - mA3