Properties of random matrices

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# Properties of random matrices
### Dr. D’Agostino McGowan

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  <span>
  Dr. Lucy D'Agostino McGowan
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## Random matrices

* `$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_

.question[
What do I mean by a _random matrix_?
]
-- 
* A _random matrix_ is a matrix consisting of a _random variable_ (as opposed to a _constant_).

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## Random matrices

* `$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_
--

* `$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_.

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## Expectation

* **Expectation** describes the _average value_
--

* In math-speak, we write it like this `$E[X]$`
--

* Recall from probability,

`$$E[X] = \begin{cases}
\sum_x xp_X(x) & X \textrm{ is a discrete random variable}\\
\int_{-\infty}^{\infty}xf_X(x)dx & X \textrm{ is a continuous random variable}
\end{cases}$$`

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## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 512 512"><path d="M256 8C119 8 8 119 8 256s111 248 248 248 248-111 248-248S393 8 256 8zm-16 328c0 8.8-7.2 16-16 16h-48c-8.8 0-16-7.2-16-16V176c0-8.8 7.2-16 16-16h48c8.8 0 16 7.2 16 16v160zm112 0c0 8.8-7.2 16-16 16h-48c-8.8 0-16-7.2-16-16V176c0-8.8 7.2-16 16-16h48c8.8 0 16 7.2 16 16v160z"/></svg> `Expectation facts`

.definition[
`$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_

`$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_
]

* `$E[\mathbf{X} + \mathbf{Y}] = E[\mathbf{X}] + E[\mathbf{Y}]$`
--

* `$E[\mathbf{AX}] = \mathbf{A}E[\mathbf{X}]$`
--

* `$E[\mathbf{A}\mathbf{X}\mathbf{B}] = \mathbf{A}E[\mathbf{X}]\mathbf{B}$`

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## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 576 512"><path d="M402.6 83.2l90.2 90.2c3.8 3.8 3.8 10 0 13.8L274.4 405.6l-92.8 10.3c-12.4 1.4-22.9-9.1-21.5-21.5l10.3-92.8L388.8 83.2c3.8-3.8 10-3.8 13.8 0zm162-22.9l-48.8-48.8c-15.2-15.2-39.9-15.2-55.2 0l-35.4 35.4c-3.8 3.8-3.8 10 0 13.8l90.2 90.2c3.8 3.8 10 3.8 13.8 0l35.4-35.4c15.2-15.3 15.2-40 0-55.2zM384 346.2V448H64V128h229.8c3.2 0 6.2-1.3 8.5-3.5l40-40c7.6-7.6 2.2-20.5-8.5-20.5H48C21.5 64 0 85.5 0 112v352c0 26.5 21.5 48 48 48h352c26.5 0 48-21.5 48-48V306.2c0-10.7-12.9-16-20.5-8.5l-40 40c-2.2 2.3-3.5 5.3-3.5 8.5z"/></svg> `Try it`

.definition[
a is a constant, `$\mathbf{X}$` and `$\mathbf{Y}$` are a random matrices
]

Solve the following:

* `$E[a\mathbf{X}]$`
* `$E[\mathbf{X} + \mathbf{Y}]$`
* `$E[\mathbf{X}\mathbf{Y}]$`

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.definition[
a is a constant, `$\mathbf{X}$` and `$\mathbf{Y}$` are a random matrices
]

Solve the following:

* `$E[a\mathbf{X}] = aE[\mathbf{X}]$`
* `$E[\mathbf{X} + \mathbf{Y}] = E[\mathbf{X}] + E[\mathbf{Y}]$`
* `$E[\mathbf{X}\mathbf{Y}]= ?$`

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## Covariance

* **Covariance** is a measure of _joint variability_ between two random variables
--

* In math-speak we write it like this: `$cov(\mathbf{X},\mathbf{Y})$`
--

* Recall from probability,

`$$cov(\mathbf{X},\mathbf{Y}) = E[(\mathbf{X} - E[\mathbf{X}])(\mathbf{Y} - E[\mathbf{Y}])]$$`

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.definition[
`$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_  
`$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_  
`$c$` and `$d$` are _constants_
]

* `$cov(\mathbf{X},\mathbf{Y}) = cov(\mathbf{Y}^T, \mathbf{X})$`
--

* `$cov(\mathbf{X} + c, \mathbf{Y} + d) = cov(\mathbf{X}, \mathbf{Y}) + cov(\mathbf{X}, d) + cov(c, \mathbf{Y}) + cov(c, d)$`
--

.question[
What is the covariance between an random variable and a constant? 
]

--
* `$cov(\mathbf{X} + c, \mathbf{Y} + d) = cov(\mathbf{X}, \mathbf{Y}) + 0 + 0 + 0 = cov(\mathbf{X}, \mathbf{Y})$`
--

* `$cov(\mathbf{A}\mathbf{X}, \mathbf{B}\mathbf{Y}) = \mathbf{A}cov(\mathbf{X}, \mathbf{Y})\mathbf{B}^T$`

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.definition[
`$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_  
`$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_  
`$c$` and `$d$` are _constants_
]

Calculate the covariance of the following:

* `$cov(\mathbf{B}\mathbf{X}, \mathbf{Y})$`
* `$cov(c, \mathbf{Y})$`

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.definition[
`$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_  
`$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_  
`$c$` and `$d$` are _constants_
]

Calculate the covariance of the following:

* `$cov(\mathbf{B}\mathbf{X}, \mathbf{Y}) = \mathbf{B}cov(\mathbf{X}, \mathbf{Y})$`
* `$cov(c, \mathbf{Y}) = 0$`

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## Variance

* **Variance** measures the spread of a variable
--

* In math-speak, we write it like this: `$var(\mathbf{X})$`
--

* Recall from probability,

`$$var(\mathbf{X}) = cov(\mathbf{X}, \mathbf{X}) = E[\mathbf{X}^2] - E[\mathbf{X}]^2$$`

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.definition[
`$\mathbf{X}$` and `$\mathbf{Y}$` are _random matrices_  
`$\mathbf{A}$` and `$\mathbf{B}$` are _constant matrices_  
`$c$` and `$d$` are _constants_
]

* `$var(\mathbf{X} + c) = var(\mathbf{X})$`
--

* `$var(\mathbf{A}\mathbf{X}) = \mathbf{A}var(\mathbf{X})\mathbf{A}^T$`
--

* If `$X_i$`s are independent, `$var(\sum_{i=1}^n X_i) = \sum var(X_i)$`

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