**3.1 Singular value decomposition (SVD)**

Singular values of the channel matrix H gives highest number of data streams which can be sent at the same time. The channel matrix H can be decomposed into a product of three matrices as follows

$$[H]\_{N\_R \times N\_T} = U \cdot \Sigma \cdot V^H \tag{6}$$

U & V are a unitary matrices with dimensions of *NR* � *NR* and *NT* � *NT* i.e.

$$UU^H = VV^H = I\_r \tag{7}$$

The VH is the transpose and complex conjugate of the matrix V, where rank of matrix H is expressed as,

$$r \le \min\left(N\_{\mathbb{R}}, N\_{\cdot T}\right) \tag{8}$$

and Σ is a *NR* � *NT* matrix with zero elements but the diagonal elements are non-zero

$$\Sigma = \operatorname{diag} \{ \sigma\_1, \sigma\_2, \dots \sigma\_r \}, \sigma\_i \ge 0 \text{ and } \sigma\_i \ge \sigma\_{i+1} \tag{9}$$

The diagonal elements are arranged in such a way that 1st element is greater than the 2nd element and so on the last element is lowest singular value and all of them are non-negative.

## **3.2 Eigen value decomposition (EVD)**

The square of singular values, *σ*<sup>2</sup> *<sup>i</sup>* , are the eigenvalues of the positive semi definite Hermitian matrix HHH. So this can be decomposed into

$$\text{HH}^H = \mathbf{Q} \cdot \boldsymbol{\Lambda} \cdot \mathbf{Q}^H \tag{10}$$

Where Q is an Orthonormal matrix with *NR* � *NR* dimension i.e.,

$$\mathbf{Q}\mathbf{Q}^H = \mathbf{Q}^H \mathbf{Q} = I \mathbf{v}\_{\cdot \mathbb{R}} \tag{11}$$

and Λ is a matrix of dimension *NR* � *NT* with zero elements but the diagonal elements are non-zero

$$\Lambda = \text{diag}\{\lambda\_1, \lambda\_2, \dots \lambda\_{N\_R}\}, \lambda\_i \ge 0 \tag{12}$$

The order of eigenvalues are *λ<sup>i</sup>* ≥ *λ<sup>i</sup>*þ1, where

$$
\lambda\_i = \begin{cases}
\sigma\_i^2 & \text{if } i = \mathbf{1}, 2, \dots \dots r \\
\mathbf{0} & \text{if } i = r+\mathbf{1}, r+\mathbf{2}, \dots \dots N\_R
\end{cases}
\tag{13}
$$

#### *3.2.1. Importance of eigenvalues*

Eigenvalue matrix of Q (*λ*<sup>1</sup> ≥*λ*<sup>2</sup> ≥ … ≥*λr*) contains the strength information about the channel i.e. *λi*. The smallest eigenvalue *λ<sup>r</sup>* is the minimum mode of the channel and it is exponentially distributed. This minimum eigenvalue plays an main role in the MIMO transceiver systems performance and useful in calculating the outage probability. It is used for antenna selection techniques in which antenna set with largest *λ<sup>r</sup>* set is chosen [6], whereas the larger eigenvalues helps in choosing maximal SNR in maximal ratio transmission [7].
