Skip to main content

2.10 Determinant and Trace of Matrix- Importance

Determinant

Determinants are defined for square matrices and it is a function that maps the matrix into a real number.Determinants are important concepts in linear algebra.It is used in the analysis and solution of system of linear equations. Determinant of a matrix is represented as $det(A)$ or $|A|$.

determinants are used for testing the invertibility.  A square matrix $A \in \mathbb{R}^{n \times n}$ is invertable if and only if $det(A) \ne 0$

For a diagonal, upper triangular and lower triangular matrix, the determinant is the product of its diagonal element

Determinant measures volume.If the two sides of the parallelogram is represented as a two columns of a matrix then the absolute value of the determinant represents volume.For a parallelepiped with three sides r,b,g then determinant of the 3x3 matrix $[ r b g ]$ is the volume of the solid.


Laplace expansion can be used to find determinant.It reduces the problem of computing the determinant of $ n \times n$ matrix into computing the determinant of  $(n-1) \times (n-1)$ matrices.

Consider a matrix $A \in \mathbb{R}^{n \times n}$ Then Laplace expansion along the column $j$ is

$det(A) =  \sum_{k=1}^{n} (-1)^{k+j} a_{kj} det(A_{k,j})$

Expansion along the row is 

$det(A) =  \sum_{k=1}^{n} (-1)^{k+j} a_{jk} det(A_{j,k})$

Here $A_{k,j} \in \mathbb{R}^(n-1) \times (n-1)$ is the submatrix of $A$ that we obtain when deleting row $k$ and column $j$.

Example 3 x 3 matrix:(Laplace Expansion to find the Determinants)


so $det(A)= -5$


import numpy as np
A=np.array([[1,2,3],[3,1,2],[0,0,1]])
detA=np.linalg.det(A)
print(" det(A)")
print(detA)
o/p
det(A) 
-5.000000000000001
Note: determinant can also be computed using Sarru's rule.

Properties of determinants
For  $A \in \mathbb{R}^{n \times n}$, the determinant exhibits the following properties.

$1.det(AB)=det(A).det(B)$
$2.det(A)=det(A^T)$
$3.det(A^{-1})=1/det(A)$
4.Similar matrices possess same determinant.So the determinant is invariant to the choice of basis of a linear mapping 
5.Adding a multiple of row/column to another does not change $det(A)$
6.Multiplication of a column/row by a scalar $k$ scales the determinant by$ k$
7.Swapping two rows/columns changes the sign of $det(A)$

We can use Gauss elimination to compute $det(A)$ by bringing $A $into row echelon form.For a triangular matrix the determinant is the product of diagonal elements. 

A square matrix $A \in \mathbb{R}^{n \times n}$ has $det(A) \ne 0$ if and only if $rank(A)=n$.In other words , A is invertible if and only if it is full rank.

Trace
The trace of a square matrix $A \in \mathbb{R}^{n \times n}$ is defined as 

$tr(A)=\sum_{i=1}^{n} a_{ii}$

ie: Trace is the sum of the diagonal elements of A

import numpy as np
A=np.array([[1,2,3],[3,1,2],[0,0,1]])
detA=np.trace(A)
print("A")
print(A)
print("Trace(A)")
print(detA)
o/p
A 
[[1 2 3]
 [3 1 2] 
 [0 0 1]] 
Trace(A) 
3

Properties of trace

$1.tr(A+B)= tr(A)+tr(B)$
$2.tr(kA)=k tr(A)$
$3.tr(I_n)=n$
$4.tr(AB)=tr(BA)$
$5.tr(ABC)=tr(BCA)$  invariant under cyclic permutation
$6.tr(xy^T)=tr(y^Tx)=y^Tx$
$7.$ If A is a Transformation matrix for a linear mapping, for a different base B
     $A=S^{-1}AS$ 
     $tr(A)=tr(S^{-1}AS)=tr(SS^{-1}A)=tr(A)$
This shows that trace is independent of the basis in linear mapping.

Characteristic Polynomial
For $\lambda \in \mathbb{R}$ and a square matrix $A \in \mathbb{R}^{n \times n}$

$P_A(\lambda)=det(A-\lambda I )=C_0+C_1\lambda+C_2\lambda^2+\cdots+C_{n-1}\lambda^{n-1}+(-1)^n\lambda^n$
The characteristic polynomial will allow us to compute eigen values and eigen vectors.

In the characteristic polynomial
   $C_0=det(A)$
  $C_{n-1}=(-1)^{n-1}tr(A)$

Example:
Let $A=\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix} $

The characteristic polynomial
$P_A(\lambda)=det(A-\lambda I )=\begin{vmatrix}
2-\lambda & 1\\
1& 2-\lambda
\end{vmatrix}=(2-\lambda)^2-1=3-4\lambda+\lambda^2$
Note that the determinant of the matrix $det(A)=3=C_0$ and the trace of the matrix $tr(A)=4=(-1)^{n-1}C_{n-1}=-4$

Finding characteristic polynomial using sympy

from sympy import Matrix
M=Matrix([[2,1],[1,2]])
M.charpoly().as_expr()

O/P
$\displaystyle \lambda^{2} - 4 \lambda + 3$

Comments

Post a Comment

Popular posts from this blog

Mathematics for Machine Learning- CST 284 - KTU Minor Notes - Dr Binu V P

  Introduction About Me Syllabus Course Outcomes and Model Question Paper Question Paper July 2021 and evaluation scheme Question Paper June 2022 and evaluation scheme Overview of Machine Learning What is Machine Learning (video) Learn the Seven Steps in Machine Learning (video) Linear Algebra in Machine Learning Module I- Linear Algebra 1.Geometry of Linear Equations (video-Gilbert Strang) 2.Elimination with Matrices (video-Gilbert Strang) 3.Solving System of equations using Gauss Elimination Method 4.Row Echelon form and Reduced Row Echelon Form -Python Code 5.Solving system of equations Python code 6. Practice problems Gauss Elimination ( contact) 7.Finding Inverse using Gauss Jordan Elimination  (video) 8.Finding Inverse using Gauss Jordan Elimination-Python code Vectors in Machine Learning- Basics 9.Vector spaces and sub spaces 10.Linear Independence 11.Linear Independence, Basis and Dimension (video) 12.Generating set basis and span 13.Rank of a Matrix 14.Linear Mapping and Matri
Post a Comment
Read more

1.1 Solving system of equations using Gauss Elimination Method

Elementary Transformations Key to solving a system of linear equations are elementary transformations that keep the solution set the same, but that transform the equation system into a simpler form: Exchange of two equations (rows in the matrix representing the system of equations) Multiplication of an equation (row) with a constant  Addition of two equations (rows) Add a scalar multiple of one row to the other. Row Echelon Form A matrix is in row-echelon form if All rows that contain only zeros are at the bottom of the matrix; correspondingly,all rows that contain at least one nonzero element are on top of rows that contain only zeros. Looking at nonzero rows only, the first nonzero number from the left pivot (also called the pivot or the leading coefficient) is always strictly to the right of the  pivot of the row above it. The row-echelon form is where the leading (first non-zero) entry of each row has only zeroes below it. These leading entries are called pivots Example: $\begin
Post a Comment
Read more

4.3 Sum Rule, Product Rule, and Bayes’ Theorem

 We think of probability theory as an extension to logical reasoning Probabilistic modeling  provides a principled foundation for designing machine learning methods. Once we have defined probability distributions corresponding to the uncertainties of the data and our problem, it turns out that there are only two fundamental rules, the sum rule and the product rule. Let $p(x,y)$ is the joint distribution of the two random variables $x, y$. The distributions $p(x)$ and $p(y)$ are the corresponding marginal distributions, and $p(y |x)$ is the conditional distribution of $y$ given $x$. Sum Rule The addition rule states the probability of two events is the sum of the probability that either will happen minus the probability that both will happen. The addition rule is: $P(A∪B)=P(A)+P(B)−P(A∩B)$ Suppose $A$ and $B$ are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols:  $P(A∩B)=0$  The addition law then simplifies to: $P(A∪B)=P(A)+P(B)$  wh
Post a Comment
Read more