Diagonalization of Impedance Matrix

Generally an impedance matrix [Z] is a complex symmetrical matrix.
It is neither a real symmetrical matrix nor a Hermitian matrix.

However a complex impedance matrix [Z] can be diagonalized.
I don't think any impedance matrix [Z] can be diagonalized.
How can I interpret whether diagonalization is possible or not physically >

Diagonalization is only possible iff there exists a complete set of eigenvectors.

Is [Z] a Hermitian matrix?

See Horn and Johnson's Matrix Analysis.

PlanarMetamaterials said:

Diagonalization is only possible iff there exists a complete set of eigenvectors.

Click to expand...

I know such very basic thing.

PlanarMetamaterials said:

Is [Z] a Hermitian matrix?

Click to expand...

No. [Z] is not a Hermitian matrix.

Here [Z]^T means transpose of [Z], [Z]⁺ means conjugate transpose of [Z].

For lossless reciprocal circuit, [Z] is a pure imaginary symmetrical matrix which is a skew Hermitian Matrix at same time.
[Z]^T==[Z], [Z]⁺==-[Z]
[Z]^T*[Z]==[Z]*[Z]^T, [Z]⁺*[Z]==[Z]*[Z]⁺
In this case, eigenvectors are real, eigenvalues are pure imaginary.
We can always diagonalize [Z] by Orthogonal Matrix. It is not by Unitary Matrix.

On the other hand, for reciprocal circuit with loss, [Z] is neither a real symmetrical matrix nor Hermitian Matrix. It is complex symmetrical matrix.
[Z]^T==[Z], [Z]⁺!=[Z]
[Z]^T*[Z]==[Z]*[Z]^T, [Z]⁺*[Z]!=[Z]*[Z]⁺
However we can still diagonalize [Z] by Orthogonal Matrx. It is not by Unitary Matrix.

Of course, if eigenvectors are not linearly independent, diagonalization is impossible.

How can I interpret whether diagonalization is possible or not physically ?

I think it may be instructive to look at diagonalization of the matrix in the first place.

If we have such a diagonalization, then: [Z]I = Iλ, with λ being the eigenvalues and I being the currents. This means that, since the impedance matrix definition is V = [Z]I, that:

V = Iλ.

I.e., there exists a certain set of currents for which all of the corresponding voltages are related by the same value (λ, or if you prefer, a modal impedance Zm).

As to what this means physically.... good question. A series of impedances connected to all ports might do it; but I can't think what purpose it might serve.

PlanarMetamaterials said:

I think it may be instructive to look at diagonalization of the matrix in the first place.

Click to expand...

Not useful at all.

This [Z] is neither Hermitian, Skew-Hermitian nor Unitary, yet it is Symmetric and Normal.
This [Z] can be diagonalized by Orthogonal Matrix, [P].


This [Z] is also neither Hermitian, Skew-Hermitian nor Unitary, yet it is Symmetric.
However [Z] is not Normal.

This [Z] still can be diagonalized by [P]
However [P] is neither Unitary nor Orthogonal Matrix.

How can I interpret ?

[P, D] = eig(Z)


(Case-1)
Z = [1, j; j, 1]
Z : Symmetric=1
Z : Hermitian=0
Z : Skew-Hermitian=0
Z : Orthogonal=0
Z : Unitary=0
Z : Normal=1

P = [0.7071,0.7071; 0.7071,-0.7071]
P : Orthogonal=1
P : Unitary=1
rank(P)=2
det(P)=-1+j*-5.88785e-017
cond(P)=1

D = [1.0000+1.0000*j,0; 0,1.0000-1.0000*j]


(Case-2)
Z = [1, j; j, 2]
Z : Symmetric=1
Z : Hermitian=0
Z : Skew-Hermitian=0
Z : Orthogonal=0
Z : Unitary=0
Z : Normal=0

P = [0.7071,-0.6124+0.3536*j; 0.6124-0.3536*j,0.7071]
P : Orthogonal=0
P : Unitary=0
rank(P)=2
det(P)=0.75+j*-0.433013
cond(P)=1.73205

D = [1.5000+0.8660*j,0; 0,1.5000-0.8660*j]


(Case-3)
Z = [1, j; j, 3]
Z : Symmetric=1
Z : Hermitian=0
Z : Skew-Hermitian=0
Z : Orthogonal=0
Z : Unitary=0
Z : Normal=0

P = [0.7071,0.7071*j; -0.7071*j,0.7071]
P : Orthogonal=0
P : Unitary=0
rank(P)=1
det(P)=2.10734e-008+j*0
cond(P)=9.49063e+007

D = [2.0000,0; 0,2.0000]

Eigenvalues are degenerated and Eigenvectors are not linearly independent in Case-3.

Just for example.

For series RC circuit, [Z] can not be diagonalized at omega=2/(R*C).

LMA said:

For series RC circuit, [Z] can not be diagonalized at omega=2/(R*C).

Click to expand...

Thanks for very good example.

Why is a diagonalization impossible in this frequency ?
What does this frequency mean?
This frequency is twice of 3-dB cutoff frequency.