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

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