Notes in Applied Qualifying Exam

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Published	02/09/2025	\(\begin{pmatrix} A & B \\ C & D \end{pmatrix}^{-1} = \begin{pmatrix} {{c2::M}} & {{c3::-MBD^{-1} }} \\ {{c4::- D^{-1}CM}} & {{c5::D^{…
Published	02/09/2025	What is the probability density function (PDF) of a multivariate normal distribution \(\mathcal{N}(\mathbf{x} \| \boldsymbol{\mu}, \boldsymbol{\Sigma})…
Published	02/09/2025	In a change of variables from \( (x, y) \) to \( (u, v)\), we have\( dx \, dy = {{c1::\| J(u, v) \| du \, dv }}\)where\( J(u,v) = {{c2::\begin{vmatrix} …
Published	02/09/2025	What is the multivariate covariance shortcut formula?
Published	02/09/2025	\[{{c1::(I + AB)^{-1} A}} = {{c2::A (I + BA)^{-1} }}\]
Published	02/09/2025	\( \text{Cov}(A \mathbf{x}) = {{c1::A \, \text{Cov}(\mathbf{x}) A^\top }}\)
Published	02/09/2025	\[{{c1::(A + BD^{-1}C)^{-1} }} = {{c3::A^{-1} - A^{-1}B}}{{c2::(D + CA^{-1}B)^{-1} }}{{c4::CA^{-1} }}\]
Published	02/09/2025	\[\text{Tr}(ABC) = {{c1::\text{Tr}(CAB)}}\]
Published	02/09/2025	\[\|AB\| = {{c1::\|A\| \|B\|}}\]
Published	02/09/2025	If A and B are matrices of size \({{c3::N \times M}}\), then \[{{c1::\left\| \mathbf{I}_N + \mathbf{A} \mathbf{B}^{\top} \right\|}} = {{c2::\left\| …
Published	02/09/2025	Given a multivariate Gaussian \( \mathbf{x} = \begin{pmatrix} \mathbf{x}_a \\ \mathbf{x}_b \end{pmatrix} \) with mean \( \boldsymbol{\mu} = \begi…
Published	02/09/2025	\[{{c1::\left\| \mathbf{I}_N + \mathbf{a} \mathbf{b}^{\top} \right\|}} = {{c2::1 + \mathbf{a}^{\top} \mathbf{b}}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{x}} \left( \mathbf{a}^{\top} \mathbf{x} \right) = {{c1::\mathbf{a}}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{x}} \left( \mathbf{x}^{\top} \mathbf{a} \right) = {{c1::\mathbf{a}}}\]
Removal Requested	02/09/2025	{{c1::\(-\mathbf{A}^{-1} \frac{\partial \mathbf{A}}{\partial x} \mathbf{A}^{-1}\)}}
Published	02/09/2025	\[\frac{\partial}{\partial x} \left( \mathbf{A}^{-1} \right) =\]
Published	02/09/2025	\[\frac{\partial}{\partial x} \ln \|\mathbf{A}\| = \]
Published	02/09/2025	\[\frac{\partial}{\partial A_{ij}} \text{Tr} (\mathbf{A} \mathbf{B}) = {{c1::B_{ji}}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{A}} \text{Tr} (\mathbf{A} \mathbf{B}) = {{c1::\mathbf{B}^{\top} }}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{A}} \text{Tr} (\mathbf{A}^{\top} \mathbf{B}) = {{c1::\mathbf{B}}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{A}} \text{Tr} (\mathbf{A}) = {{c1::\mathbf{I}}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{A}} \text{Tr} (\mathbf{A} \mathbf{B} \mathbf{A}^{\top}) = {{c1::\mathbf{A} (\mathbf{B} + \mathbf{B}^{\top})}}\]
Published	02/09/2025	\[\frac{\partial}{\partial \mathbf{A}} \ln \|\mathbf{A}\| = {{c1::(\mathbf{A}^{-1})^{\top} }}\]
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