Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?

In summary, the conversation discusses the possibility of taking a non-diagonal Hermitian matrix A and diagonalizing it using a unitary matrix U, where U is not necessarily formed by the eigenvectors of A. The identity matrix is not a valid case for this scenario, as the eigenvalues of U and A must be different from 1 and -1. An example is given for a 2x2 matrix with eigenvalues of U being ±1, and D having non-1/-1 eigenvalues in order to satisfy the conditions.
  • #1
LagrangeEuler
717
20
Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it
[tex]UAU=D[/tex]
but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.
 
Physics news on Phys.org
  • #2
[tex]U=U^{-1}[/tex]
I am not sure of such a specific case.
 
  • #3
Yes here I am talking about case ##U=U^{-1}##. I am also not sure. But for me it is interesting.
 
  • #5
As an example in 2X2 marix
[tex]U=
\begin{pmatrix}
k & \alpha \\
\beta & -k \\
\end{pmatrix}
[/tex]
where
[tex]\alpha \beta = 1-k^2[/tex]
would produce non diagonal matrix A from diagonal matrix D which has two different eigenvalues with A=UDU.
 
Last edited:
  • #6
You can give me 2x2 example. But specify ##A##, ##U## and ##D##. Because ##U## still possibly can be formed of eigenvectors of ##A## in your example of ##U##.
 
  • #7
Eingenvalues of U are ##\pm 1##. You can pick up the cases that eigenvalues of D and A are not 1 nor -1 for your purpose.
 
  • #8
jedishrfu said:
is the identity m
No, because the identity matrix is diagonal.
 
  • #9
anuttarasammyak said:
Eingenvalues of U are ##\pm 1##. You can pick up the cases that eigenvalues of D and A are not 1 nor -1 for your purpose.
Why? Eigenvalue can be for instance ##i##.
 
  • #10
My #7 talks abot U of post #5. There for U [tex]\lambda^2 =1[/tex]
[tex]UDU=
\begin{pmatrix}
k & \alpha \\
\beta & -k \\
\end{pmatrix}
\begin{pmatrix}
d_1 & 0 \\
0 & d_2 \\
\end{pmatrix}
\begin{pmatrix}
k & \alpha \\
\beta & -k \\
\end{pmatrix}
=
\begin{pmatrix}
d_2+(d_1-d_2)k^2 & k\alpha (d_1-d_2) \\
k \beta (d_1-d_2) & d_1-(d_1-d_2)k^2 \\
\end{pmatrix}
=A[/tex]
, and
[tex]d_1,d_2 \neq -1,1[/tex]
to satisfy your further condition.
 
Last edited:

Similar threads

I Why Use Gram-Schmidt to Make a Unitary Matrix?
    • Linear and Abstract Algebra
Replies
14
Views
977
I Spectral theorem for Hermitian matrices-- special cases
    • Linear and Abstract Algebra
Replies
2
Views
465
I General worked out solution for diagonalizing a 4x4 Hermitian matrix
    • Linear and Abstract Algebra
Replies
6
Views
2K
I Change of basis matrix for point group C3V
    • Linear and Abstract Algebra
Replies
2
Views
147
I Proving U=eiA is Unitary: Exploring -1 as Exponent and Inverse
    • Linear and Abstract Algebra
Replies
7
Views
1K
How can a diagonalising matrix be unitary?
    • Advanced Physics Homework Help
Replies
15
Views
1K
A Similarity transformation changing the determinant to 1
    • Linear and Abstract Algebra
Replies
20
Views
863
I Proving only 1 normalized unitary vector for normal matrix
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Does Axler's Spectral Theorem Imply Normal Matrices?
    • Linear and Abstract Algebra
Replies
3
Views
873
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
    • Linear and Abstract Algebra
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top