# CATEGORY / Learning

### Matrix Inverse & Transpose

Matrix inverse: If is an matrix, and if it has an inverse, then

Note: Matrices that do not have an inverse are singular or degenerate.

Matrix transpose: Let be an matrix, and let . then B is an matrix and .

### Properties Of Matrix Multiplication

1. Not commutative.
2. Associative.

e.g. For where is matrix and is matrix,
is an matrix,
is an matrix.

Identity matrix
Denoted as or
e.g.

For any matrix ,

### Matrix Multiplication

3 by 2 matrix 2 by 1 matrix 3 by 1 matrix

by matrix by matrix by matrix

### Addition & Scalar Multiplication Of Matrices

Scalar multiplication:

### Matrices & Vectors

Matrix
Matrix: rectangular array of numbers
Dimension of matrix: number of rows number of columns
: , entry in the row, column

e.g.

dimension: or

Vector
Vector: matrix
: element

e.g.

dimension: 3-dimensional vector or

1-indexed vector:

0-indexed vector:

repeat until convergence {
(for and )
}

: learning rate
: assigning to

Simultaneous update
temp0 :=
temp1 :=
:= temp0
:= temp1

repeat until convergence {

}

### P-Value

p-value: probability of observing an outcome which is at least as hostile (or adversarial) to the null hypothesis as the one observed

Example
Null hypothesis: mean lifetime of a manufacturing device = 9.4 years
Accepted: within 0.396 units

50 elements with sample mean of 8.96
What is the probability that when we generate a different and independent sample average of 50 observations, we get the value <8.96 if the null hypothesis is true?

Worse than 8.96
1. Getting a number smaller than 8.96
2. Getting a number larger than 9.84

Conclusion: the larger the p-value, the stronger the evidence supporting the hypothesis.

### Validity Of Binomial Distribution

Binomial distribution: discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability

Null hypothesis: there is no there is no significant difference between specified populations, any observed difference being due to sampling or experimental error

### Hypothesis Testing

Hypothesis testing: using a data observed from a distribution with unknown parameters, we hypothesise that the parameters of this distribution take particular values and test the validity of this hypothesis using statistical methods

Confidence intervals: provide probabilistic level of certainty regarding parameters of a distribution

Example:
1.
2. unknown mean value
3. known

normal distribution:
estimate of :
distribution of :

Suppose:

### Types Of Errors

Precision: how often a classifier is right when it says something is fraud
Recall: how much of the actual fraud that we correctly detect

Harmonic mean of and

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