| Notation . | Description . |
|---|---|
| |$\mathcal {S}_{N} = \left\lbrace r_1, \dots , r_N \right\rbrace$| | Set of N random seeds rn of probability space |
| |$\boldsymbol{y}(r_n)\equiv \boldsymbol{y}_n$| | Random column vector of size p at seed rn |
| |$\mathbb {E}\left[ \boldsymbol{y} \right]\equiv \boldsymbol{\mu _y}$| | Expectation value of random vector |$\boldsymbol{y}$| |
| |$[[ m,n ]]$| | Set of integers from m to n |
| |$\boldsymbol{M}^{\boldsymbol{T}}$| | Transpose of real matrix |$\boldsymbol{M}$| |
| |$\boldsymbol{M}^{\boldsymbol{\dagger }}$| | Moore–Penrose pseudo-inverse of matrix |$\boldsymbol{M}$| |
| |$\det \left(\boldsymbol{M}\right)$| | Determinant of matrix |$\boldsymbol{M}$| |
| |$\mathbb {E} \left[\left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right]\right) \left(\boldsymbol{x} - \mathbb {E} \left[\boldsymbol{x}\right]\right)^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{xx}}}$| | Variance-covariance matrix of random vector |$\boldsymbol{x}$| |
| |$\mathbb {E} \left[ \left(\boldsymbol{y} - \mathbb {E} \left[ \boldsymbol{y} \right] \right) \left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right] \right) ^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{yx}}}$| | Cross-covariance matrix of random vectors |$\boldsymbol{y}$| and |$\boldsymbol{x}$| |
| |$\sigma _{y}^2$| | Variance of scalar random variable y |
| |$\boldsymbol{0}_{p,q}$| and |$\boldsymbol{0}_p$| | Null matrix in |$\mathbb {R}^{p \times q}$| and null vector in |$\mathbb {R}^{p}$| |
| |$\boldsymbol{I}_p$| | Square p × p identity matrix |
| Notation . | Description . |
|---|---|
| |$\mathcal {S}_{N} = \left\lbrace r_1, \dots , r_N \right\rbrace$| | Set of N random seeds rn of probability space |
| |$\boldsymbol{y}(r_n)\equiv \boldsymbol{y}_n$| | Random column vector of size p at seed rn |
| |$\mathbb {E}\left[ \boldsymbol{y} \right]\equiv \boldsymbol{\mu _y}$| | Expectation value of random vector |$\boldsymbol{y}$| |
| |$[[ m,n ]]$| | Set of integers from m to n |
| |$\boldsymbol{M}^{\boldsymbol{T}}$| | Transpose of real matrix |$\boldsymbol{M}$| |
| |$\boldsymbol{M}^{\boldsymbol{\dagger }}$| | Moore–Penrose pseudo-inverse of matrix |$\boldsymbol{M}$| |
| |$\det \left(\boldsymbol{M}\right)$| | Determinant of matrix |$\boldsymbol{M}$| |
| |$\mathbb {E} \left[\left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right]\right) \left(\boldsymbol{x} - \mathbb {E} \left[\boldsymbol{x}\right]\right)^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{xx}}}$| | Variance-covariance matrix of random vector |$\boldsymbol{x}$| |
| |$\mathbb {E} \left[ \left(\boldsymbol{y} - \mathbb {E} \left[ \boldsymbol{y} \right] \right) \left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right] \right) ^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{yx}}}$| | Cross-covariance matrix of random vectors |$\boldsymbol{y}$| and |$\boldsymbol{x}$| |
| |$\sigma _{y}^2$| | Variance of scalar random variable y |
| |$\boldsymbol{0}_{p,q}$| and |$\boldsymbol{0}_p$| | Null matrix in |$\mathbb {R}^{p \times q}$| and null vector in |$\mathbb {R}^{p}$| |
| |$\boldsymbol{I}_p$| | Square p × p identity matrix |
| Notation . | Description . |
|---|---|
| |$\mathcal {S}_{N} = \left\lbrace r_1, \dots , r_N \right\rbrace$| | Set of N random seeds rn of probability space |
| |$\boldsymbol{y}(r_n)\equiv \boldsymbol{y}_n$| | Random column vector of size p at seed rn |
| |$\mathbb {E}\left[ \boldsymbol{y} \right]\equiv \boldsymbol{\mu _y}$| | Expectation value of random vector |$\boldsymbol{y}$| |
| |$[[ m,n ]]$| | Set of integers from m to n |
| |$\boldsymbol{M}^{\boldsymbol{T}}$| | Transpose of real matrix |$\boldsymbol{M}$| |
| |$\boldsymbol{M}^{\boldsymbol{\dagger }}$| | Moore–Penrose pseudo-inverse of matrix |$\boldsymbol{M}$| |
| |$\det \left(\boldsymbol{M}\right)$| | Determinant of matrix |$\boldsymbol{M}$| |
| |$\mathbb {E} \left[\left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right]\right) \left(\boldsymbol{x} - \mathbb {E} \left[\boldsymbol{x}\right]\right)^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{xx}}}$| | Variance-covariance matrix of random vector |$\boldsymbol{x}$| |
| |$\mathbb {E} \left[ \left(\boldsymbol{y} - \mathbb {E} \left[ \boldsymbol{y} \right] \right) \left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right] \right) ^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{yx}}}$| | Cross-covariance matrix of random vectors |$\boldsymbol{y}$| and |$\boldsymbol{x}$| |
| |$\sigma _{y}^2$| | Variance of scalar random variable y |
| |$\boldsymbol{0}_{p,q}$| and |$\boldsymbol{0}_p$| | Null matrix in |$\mathbb {R}^{p \times q}$| and null vector in |$\mathbb {R}^{p}$| |
| |$\boldsymbol{I}_p$| | Square p × p identity matrix |
| Notation . | Description . |
|---|---|
| |$\mathcal {S}_{N} = \left\lbrace r_1, \dots , r_N \right\rbrace$| | Set of N random seeds rn of probability space |
| |$\boldsymbol{y}(r_n)\equiv \boldsymbol{y}_n$| | Random column vector of size p at seed rn |
| |$\mathbb {E}\left[ \boldsymbol{y} \right]\equiv \boldsymbol{\mu _y}$| | Expectation value of random vector |$\boldsymbol{y}$| |
| |$[[ m,n ]]$| | Set of integers from m to n |
| |$\boldsymbol{M}^{\boldsymbol{T}}$| | Transpose of real matrix |$\boldsymbol{M}$| |
| |$\boldsymbol{M}^{\boldsymbol{\dagger }}$| | Moore–Penrose pseudo-inverse of matrix |$\boldsymbol{M}$| |
| |$\det \left(\boldsymbol{M}\right)$| | Determinant of matrix |$\boldsymbol{M}$| |
| |$\mathbb {E} \left[\left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right]\right) \left(\boldsymbol{x} - \mathbb {E} \left[\boldsymbol{x}\right]\right)^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{xx}}}$| | Variance-covariance matrix of random vector |$\boldsymbol{x}$| |
| |$\mathbb {E} \left[ \left(\boldsymbol{y} - \mathbb {E} \left[ \boldsymbol{y} \right] \right) \left(\boldsymbol{x} - \mathbb {E} \left[ \boldsymbol{x}\right] \right) ^{\boldsymbol{T}} \right]\, \equiv \boldsymbol{\Sigma _{\boldsymbol{yx}}}$| | Cross-covariance matrix of random vectors |$\boldsymbol{y}$| and |$\boldsymbol{x}$| |
| |$\sigma _{y}^2$| | Variance of scalar random variable y |
| |$\boldsymbol{0}_{p,q}$| and |$\boldsymbol{0}_p$| | Null matrix in |$\mathbb {R}^{p \times q}$| and null vector in |$\mathbb {R}^{p}$| |
| |$\boldsymbol{I}_p$| | Square p × p identity matrix |
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