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Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to zeroinfinity with fixed ratio)? \begin{align} \mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}, \end{align} as $\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$, where $\mathbf{I}_m$ is an identity matrix, we have \begin{align} \frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &\rightarrow \frac{n}{m}\mathbf{I}_n. \end{align}

Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to zero with fixed ratio)? \begin{align} \mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}, \end{align} as $\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$, where $\mathbf{I}_m$ is an identity matrix, we have \begin{align} \frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &\rightarrow \frac{n}{m}\mathbf{I}_n. \end{align}

Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with fixed ratio)? \begin{align} \mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}, \end{align} as $\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$, where $\mathbf{I}_m$ is an identity matrix, we have \begin{align} \frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &\rightarrow \frac{n}{m}\mathbf{I}_n. \end{align}

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Large scale analysis of matrix multiplications

Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to zero with fixed ratio)? \begin{align} \mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}, \end{align} as $\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$, where $\mathbf{I}_m$ is an identity matrix, we have \begin{align} \frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\ &\rightarrow \frac{n}{m}\mathbf{I}_n. \end{align}