# How Does the Covariance Matrix Apply to Vectors X and Y?

• DUET
In summary, the 3x3 covariance matrix for X and Y would have two diagonal elements (var(X) and var(Y)), and two off diagonal elements (cov(X,Y) and cov(Y,X)). The difference between #1 and #2 is that #1 is for real valued random variables, while #2 is a generalization for vectors with random variables as components.
DUET
if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?

DUET said:
if X= (3, 5, 7) & Y = (2, 4, 1)

What is the 3x3 covariance matrix for X & Y?

You have two variables, so the matrix is 2x2. The elements are var(X), var(Y) along the diagonal and cov(X,Y) off diagonal (both).

Since 2x2 we need two diagonal elements and two off diagonal elements.

Are the following two elements "off diagonal elements"?

cov(X,Y) & cov(Y,X);

Last edited:
Yes they are the off diagonal elements.

The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?

Last edited:
DUET said:
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?

In this context what do you mean by dimensional? X and Y are real valued. Do you mean the number of samples?

DUET said:
The covariance between two jointly distributed real-valued random variables x and y with finite second moments is defined as-
1. cov(x,y)=E[(x-E[x])(y-E[y])]

The covariance between two jointly distributed real-valued random vectors x and y (with m and n dimensional respectively) with finite second moments is defined as
2. cov(x,y)=E[(x-E[x])(y-E[y])T]

What is the difference between #1 & #2?

1 refers to real valued (1 dimensional) random variables.
2 is a generalization to vectors (n or m dimensional) which have random variables as components.

## What is a covariance matrix?

A covariance matrix is a mathematical matrix that contains the variances and covariances between multiple variables. It is used to describe the linear relationship between two or more variables in a dataset.

## How is a covariance matrix calculated?

A covariance matrix is calculated by taking the variance of each variable and the covariance between each pair of variables in a dataset. The resulting matrix will have the same number of rows and columns as the number of variables in the dataset.

## What does a positive covariance in a covariance matrix indicate?

A positive covariance in a covariance matrix indicates that the two variables have a positive linear relationship. This means that when one variable increases, the other variable also tends to increase.

## What does a negative covariance in a covariance matrix indicate?

A negative covariance in a covariance matrix indicates that the two variables have a negative linear relationship. This means that when one variable increases, the other variable tends to decrease.

## Why is the covariance matrix important?

The covariance matrix is important because it provides valuable information about the relationship between variables in a dataset. It is often used in statistical analysis and machine learning algorithms to understand the variability and dependencies between variables.

Replies
4
Views
2K
Replies
6
Views
1K
Replies
10
Views
720
Replies
1
Views
8K
Replies
24
Views
2K
Replies
1
Views
322
Replies
2
Views
2K
Replies
14
Views
451
Replies
12
Views
4K
Replies
15
Views
5K