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Suppose that $A_{2\times 2}$ is a hermitian matrix, so it has real eigenvalues $\lambda_1$ and $\lambda_2$ and corresponding orthonormal eigenvectors $\underline u_1$ and $\underline u_2$ and we know it can be written as the sum of two $rank-1$ hermitian matrices: $\text{T is the symbol for transpose and * for complex conjugate}$ $$A=\lambda_1\underline u_1\otimes\underline u_1+\lambda_2\underline u_2\otimes\underline u_2=\lambda_1\underline u_1\underline u_1^{*T}+\lambda_2\underline u_2\underline u_2^{*T}\qquad $$
Both $\underline u_1\otimes \underline u_1$ and $\underline u_2\otimes \underline u_2$ are hermitian so $(\underline u_1\otimes \underline u_1+ \underline u_2\otimes \underline u_2)$ is hermitian. In a book on page 50, it says that:

As the two unit orthogonal eigenvectors verify $\underline u_1\underline u_1^{*T}+\underline u_2 \underline u_2^{*T}=I_{D2}$, it follows the Chandrasekhar decomposition of the wave given by $$J=(\lambda_1-\lambda_2)\underline u_1\underline u_1^{*T}+\lambda_2I_{D2}=J_{CP}+J_{CD}$$

My question is what do we mean by $I_{D2}$ matrix?
Also what is Chandrasekhar decomposition? Can you introduce me online resources to study?
Or can you guide me to the Chanrasekhar's original paper in which he introduced this decomposition?

Greenonline
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Sepideh Abadpour
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1 Answers1

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From the general properties of 2x2 matrices, if ${\underline u}_1$, ${\underline u}_2$ are the eigenvectors of $A_{2x2}$, then $$ {\underline u}_1 {\underline u}^{*T}_1 + {\underline u}_2 {\underline u}^{*T}_2 = I_{2x2} $$

As for the Chandrasekhar decomposition, perhaps the explanation on pgs.269-271 in "Direct and Inverse Methods in Radar Polarimetry" (Google Books link) can help. It seems it is a decomposition of the coherency matrix J into a pure state (perfectly polarized) component, $(\lambda_1 - \lambda_2){\underline u}_1 {\underline u}^{*T}_1$, and a randomly polarized, noise term, $\lambda_2 I_{2x2}$.

udrv
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  • @urdu $I_{2x2}$ is the identity $2\times 2$ matrix? Is this true just for $2\times 2$ matrices? Because in my math class for a hermitian $3\times 3$ matrix I had the following result for the aformentioned sum: $$\begin{bmatrix}5&10&0\10&25&5\0&5&5\end{bmatrix}$$ – Sepideh Abadpour Aug 18 '15 at 04:34
  • Consider the matrix $$\begin{bmatrix}-1&2\1&0\end{bmatrix}$$, Its eigenvalues are $\lambda_1=1$ and $\lambda_2=-2$ so its eigenvectors will be $$\bagin{bmatrix}1\1\end{bmatrix}$$ and $$\bagin{bmatrix}-2\1\end{bmatrix}$$ and we will have:$$\bagin{bmatrix}1\1\end{bmatrix}\bagin{bmatrix}1&1\end{bmatrix}+\bagin{bmatrix}-2\1\end{bmatrix}\bagin{bmatrix}-2&1\end{bmatrix}=\begin{bmatrix}5&-1\-1&2\end{bmatrix}$$ so if you mean an identity 2-by-2 matrix from $I+{2\times 2}$, it's not always verified – Sepideh Abadpour Aug 18 '15 at 05:03
  • Consider the matrix $$\begin{bmatrix}-1&2\1&0\end{bmatrix}$$, Its eigenvalues are $\lambda_1=1$ and $\lambda_2=-2$ so its eigenvectors will be $\begin{bmatrix}1\1\end{bmatrix}$ and $\begin{bmatrix}-2\1\end{bmatrix}$ and we will have:$$\begin{bmatrix}1\1\end{bmatrix}\begin{bmatrix}1&1\end{bmatrix}+\begin{bmatrix}-2\1\end{bmatrix}\begin{bmatrix}-2&1\end{bmatrix}=\begin{bmatrix}5&-1\-1&2\end{bmatrix}$$ so if you mean an identity 2-by-2 matrix from $I_{2\times 2}$, it's not always verified – Sepideh Abadpour Aug 18 '15 at 05:16
  • Oh yes maybe it is verified for $2\times 2$ hermitian matrices. I should prove it – Sepideh Abadpour Aug 18 '15 at 05:53
  • @sepideh Yes, I_2x2 is the 2x2 identity matrix. The identity for the eigenvectors holds only for hermitian 2x2 matrices. It does hold for nxn matrices as well if you extend the sum to include all eigenvectors. For non-hermitian matrices it involves left-eigenvectors instead of the complex transposed ones. – udrv Aug 18 '15 at 07:58
  • @but I'm sure it does not hold for nxn hermitian matrices. – Sepideh Abadpour Aug 18 '15 at 08:06
  • Can you suggest me the book where this property of hermitian matrices is introduced or proved? – Sepideh Abadpour Aug 18 '15 at 08:08
  • I understood the proof http://math.stackexchange.com/questions/1401253/prove-that-the-identity-sum-i-1n-underline-u-i-underline-u-it-i-n-time , you are right so the answer to the second part of my question is to study the pages 269-271 of the suggested book – Sepideh Abadpour Aug 18 '15 at 10:54