Matrix
Before we try to understand Matrix we should get familiar with Vectors as Matrix is simply multiple vectors put together.
The article only focuses on the the concepts required for the data science and not every concepts of matrices.
Matrix
A Matrix represents a set of vectors put together. Each column of a matrix represents a vector and each row represents the elements of the vector,
Representation
A Matrix is represented by a capital letter, A — Z. The elements in the matrix is represented by lowercase letters, a — z, with number of rows followed by number of columns.
Types of Matrix
Column Matrix
This is simply a matrix with just 1 column.
Row Matrix
This is simply a matrix with just 1 row.
Null or Zero Matrix
All the elements of the matrix are equal to 0.
Square Matrix
A matrix with equal number of rows and columns.
Diagonal Matrix
All elements other than diagonal are equal to 0.
A diagonal matrix with all the values in diagonal equal to 1 is called Unit or Identity matrix.
There are few other types of matrices such as scalar matrix, upper triangular matrix, lower triangular matrix, symmetric matrix, and antisymmetric matrix but we do not need to focus too much on those.
Operations
Addition
We can only add 2 or more matrices if the number of rows and columns of both the matrices are the same.
Subtraction
We can only subtract 2 or more matrices if the number of rows and columns of both the matrices are the same.
Multiplication
Scalar Multiplication
Every element of a matrix is multiplied by the scalar value.
Properties
Here are some of the properties of scalar multiplication.
Matrices Multiplication
The multiplication is only possible when the number of columns in the first matrix and rows in the second matrix are equal.
Each element of a row from first matrix is multiplied by the each element of the column of the second matrix.
This is also known as Dot Product for matrices.
Properties
Here are some of the properties of matrices multiplication.
Transpose
We replace the rows of a matrix to the columns and columns to the rows. Interchanging of rows and columns is known as the transpose of matrices.
Properties
Here are some of the properties of matrices transpose.
Determinant
Determinant gives us the scalar value and it can only be calculated for a square matrix.
Note the sequence of +,-,+… while calculating the determinant.
Inverse
The inverse of a square matrix, sometimes called a reciprocal matrix, is a matrix such that.
To calculate the inverse of a matrix we need to understand the Minor, Cofactor and Adjoin of matrices. For the purpose of data science we don’t need to go into such details.
Hadamard Product
Hadamard product or element-wise product takes two matrices of the same dimensions and produces another matrix of the same dimension.
Echelon Form
A matrix is in Echelon form if it satisfies the below properties.
- Any rows, if any, of all zeroes is the last row.
2. Each leading entry of a row is in a column to the right of the leading entry in the row above it.
3. All entries in a column below a leading entry are zero.
Row (Column) Echelon Form
Apart from the properties above there is one additional important property here i.e. the first non-zero element in each row (column), called the leading entry, is 1.
Reduced Row (column) Echelon Form
Apart from the properties above here is one additional important property here i.e. the leading entry in each row (column) is the only non-zero entry in its column (row).
Rank
Rank is one of the important concept in Matrix. It helps us identify linearity in the matrix. The Rank tells us the maximum number of linearly independent columns or rows in a matrix.
To calculate the rank we first check if the determinant of the matrix is 0 or not.
- If |A| != 0, then the rank of the matrix is the number of rows of the matrix.
- If |A| = 0, we convert the matrix into its echelon form by reducing the rows until it cannot be further.
Example
|A| != 0
|A| = 0
Now we apply elementary transformations (row reduce)
We get
Again,
We get
We cannot further reduce the matrix so we stop here. The number of non — zero rows is 2. So the rank of the matrix is 2.
Linear Independence
Set of vectors are independent if they do not lie on the same plane. If the vectors lie on the same plane they are dependent.
A set of vectors {v1, v2, …, vk} is linearly independent if the vector equation
has only the trivial solution
The set {v1, v2, …, vk} is linearly dependent otherwise.
Example
Linear Dependence
Given Vectors
Vector Equation
We solve this by forming a matrix and row reducing
We get,
And finally,
So there exist non-trivial solutions. For instance, taking z=1 (!= 0) gives this equation of linear dependence.
So, there is a solution other than the trivial solution. We conclude that the set is Linearly Dependent.
Linear Independence
Given Vectors
Vector Equation
We solve this by forming a matrix and row reducing
We get,
And finally,
So, the only solution is the trivial solution. We conclude that the set is Linearly Independent.
I hope this article provides you with a good understanding of some important concepts of Matrices.
If you have any questions or if you find anything misrepresented please let me know.
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