Basic Concepts, Matrix Addition & Scalar Multiplication | Mathematical Methods - Physics PDF Download

Basic Concepts

A matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. Thus a m x n matrix is of the form
 
In matrix A has m rows and n column a_ij is   the element of i^th row and j^th column.
If number of row and number of column in matrix is equal (m = n) then matrix is said to be square matrix. Then its diagonal containing the entries a₁₁ ,a₂₂ , a_nn ...., ann is called main diagonal or principal diagonal of A.

A matrix that is not diagonal is called a rectangular matrix.
Transposition
 A transpose A^T of a m x n matrix A =[a_ij] is the n x m matrix that has the first row of A as its first column, the second row of A as its second column, and so on. Thus the transpose of A is

Symmetric matrices and skew-symmetric matrices
Matrices whose transpose equals the matrix are called symmetric matrix. Matrices whose transpose equals the minus of matrix are called anti-symmetric matrix
A^T = A (symmetric matrix)  and A^T = -A (anti-symmetric matrix)
Equality of matrices
Two matrices A [a_ij] and B [b_ij] are equal, written A = B , if and only if they have the same size and the corresponding entries are equal, that is a₁₁ = b₁₁ , a₁₂ = b₁₂ and so on.

Matrix Addition

Addition is defined only for matrices A [a_ij] and B = [b_ij] of the same size; their sum written A + B , is then obtained by adding the corresponding entries. Matrices of different sizes cannot be added

Scalar Multiplication

The product of any m x n matrix A = [a_ij]  and any scalar λ =  written λ A is the m x n matrix λA =[λa_ij] obtained by multiplying each entry in A by λ. Thus

Here (-1) A is simply written -A and is called the negative of A . Similarly, (-k)A is simply written -kA . Also, A +(-B) is written A - B and is called the difference of A and B (which must have the same size).
For matrices of the same size m x n we obtain for addition
(i) A + B = B + A
(ii) (A + B) + C = A + (B + C)

(iii) A + 0 = A
(iv) A + (-A) = 0 here 0 denote the zero matrix of size m x n.
and for scalar multiplication

(i) λ(A + B) = λA + λB

(ii) (λ + k )A = λA + kA

(iii) λ(kA) = (λk )A

(iv) 1A = A
Transposition of sum can be done term by term (A + B)^T =A^T + B^T and for scalar multiplication we have (λA)^T = λA^T

Matrix Multiplication

The product C = AB (in this order) of an m x n matrix A = [a_ij] and r x p matrix B = [b_ij] is defined if and only if r = n, that is
Number of rows of 2^nd factor B= Number of columns of 1^st factor A
and is then defined as the m x p matrix C = [c_ij] with entries

 Properties of Matrix Multiplication

(i) AB ≠ BA in general

(ii) AB = 0 does not necessarily imply A = 0 or B = 0 or BA = 0

(iii) AC = AD does not necessarily imply C = D .

(iv) (kA)B = k(AB) = A(kB)

(v) A(BC) = (AB)C
(vi) (A + B )C = AC + BC

(vii) (AB)^T = B^TA^T

Special Matrices

Triangular Matrices

Upper triangular matrices are square matrices that can have nonzero entries only on and above the main diagonal, whereas any entry below the diagonal must be zero.

Similarly lower triangular matrices can have nonzero entries only on and below the main diagonal, whereas any entry above the diagonal must be zero.  Any entry on the main diagonal of a triangular matrix may be zero or not
 

Diagonal Matrices

These are square matrices that can have nonzero entries only on the main diagonal. Any entry above or below the main diagonal must be zero.

Scalar Matrix

If all the diagonal entries of a diagonal matrix S are equal, say, c , we call S a scalar matrix because multiplication of any square matrix A of the same size by S has same effect as the multiplication by scalar, that is,
AS = SA = cA

Example:

and

Unit Matrix (Identity Matrix)

A scalar matrix whose entries on the main diagonal are all 1 is called a unit matrix (identity matrix) and is denoted by I . Thus
AI = IA = A

Inverse of a Matrix

The inverse of a n x n matrix A = [aij] is denoted by A^-1 and is an n
n matrix such that AA^-1 = A^-1 A = I
where I is the n x n unit matrix.

Note:

(i) If A has inverse, then A is called a nonsingular matrix.

(ii) If A has no inverse, then A is called a singular matrix.

(iii) If A has an inverse, the inverse is unique.
The inverse of a nonsingular n x n matrix  A = [aij] is given by

where A_ij is the cofactor of a_ij in det A . Note well that in A^-1, the cofactor A_ij occupies the same place as a_ji does in A .

Cofactor of a<sub>ij
A determinant of order n is a scalar associated with an n
n matrix A = [aij], which is written

and is defined for n = 1 by  D = a11 ,
and is defined for n > 2 by</sub>
D = a_i1 C_i1 + a_{i2Ci2 + •••• +} a_{inCin    (i = 1,2 .......,or n)}
where C_ij = (-1_i)^{i + j} M_ij
and M_ij is a determinant of order (n - 1), namely, the determinant of the submatrix of A obtained from A by deleting the row and column of the entry a_ij .
M_ij is called the minor of a_ij in D and C_ij is the cofactor of a_ij in D .
Example 1:  Let

Example 2: Let then cofactor of A is 

Hence,

Example 3: Let

Matrix Eigen value Problems

Let A = [a_ij] be a given n xn square matrix and consider the equation
AX = λX ........(1)
Here X is an unknown vector and λ an unknown scalar and, we want to determine both.

A value of λ for which (1) has a solution X ≠ 0 is called eigenvalue of the matrix A . The corresponding solutions X ≠ 0 of (1) are called eigenvevtors of A corresponding to that eigenvalue A.

In matrix notation,    (A-λI)X = 0  ........(2)
This homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero

D (λ) is called the characteristic determinant. The equation is called the characteristic

equation of the matrix A . By developing D (λ) we obtain a polynomial of nth degree

in λ . This is called the characteristic polynomial of A .

Note:

• The eigenvalues of a square matrix A are the roots of the characteristic equation (3) of A . Hence an n x n matrix has at least one eigenvalue and at most n numerically different eigenvalues.
• Once the eigenvalues are known, corresponding eigenvectors are obtained.
• Repeated eigenvalues are said to be degenerate eigenvalues. For degenerate eigenvalues there are different eigenvectors for same eigenvalues.
• Non repeated eigenvalues are non-degenerate eigenvalues. For non-degenerate eigenvalues there are different eigenvectors for different eigenvalues.
• Sum of eigenvalues are equal to trace of matrix Trace of matrix is sum of diagonal element.
• Product of eigenvalues are equal to determinant of matrix
• Eigenvectors correspond to different eigenvalues are always independent.
• Eigenvectors corresponds to same eigenvalue may or may not be independent.

Orthonormality Condition

If i = j then δ_ij = 1 (Normalisation Condition)
and If i ≠ j then δ_ij = 0 (Orthogonal Condition)

Linear independence and dimensionality of a vector space
A set of A vectors X₁,X₂,...,X_N is said to be linearly independent if

is satisfied when a₁ = a₂ = a₃ = a₄ =........= 0 otherwise it is said to be linear dependent.

The dimension of a space vector is given by the maximum number of linearly independent vectors the space can have.
The maximum number of linearly independent vectors a space has isN(X₁X₂,...,X_N) . This space is said to be N dimensional. In this case any vector Y of the vector space can be expressed as linear combination.

Example 4: Find the eigenvalues and orthonormal eigenvectors of matrix

For eigenvalues
(2-λ) {X2 - 1} = 0 ⇒ λ = 2 and λ = +1
Eigenvalues are given by λ =-1, X₂ = 1, λ3 = 2. All eigenvalues are different so they are non-degenerate.

Eigenvector can be determined by the equation AX = λX .

For λ1 = -1

Normalized eigenvector can be determined by relation Xx = 1.

Normalized Eigenvector corresponds to λ = -1 is
For λ₂ = 1

Normalized eigenvector can be determined by relation X₂^T X₂ = 1 ⇒ X₂ =

For  λ3 = 2

Normalized eigenvector can be determined by relation X₃^TX₃ = 1

Example 5: Find the eigenvalues and orthonormal eigenvectors of matrix A =

Also check that eigenvectors are linearly independent.

For eigenvalues (A - λI) = 0
(1-λ){λ²-l} = 0 ⇒ 1 = 1, 1,-1
where λ = -1 is non-degenerate eigenvalue and λ₂ = λ₃ = 1 is doubly degenerate eigenvalue.
Eigenvector can be determined by the equation AX = λX .

For λ₁ =-1

Normalized eigenvector can be determined by relation
For λ₂, = λ₃ = 1

So eigenvector is given byfor any value of x, and x,. One can find Eigenvector corresponds to λ₂ = λ₃ = 1 but we need to find orthogonal Eigenvectors.
Let X₁ = X₁ and x₂ = 0 so X₂ =
Normalized eigenvector can be determined by relation X₂^TX₂, =1 ⇒ X2 =
Now we must find third Eigenvector which will satisfied equation x₁ = x₁ and x₂ = x₃ and orthogonal to X₂.

So and value of x₂ can be find with orthogonal relation X₃^TX₃ = 1



⇒ a₁ = 0, a₂ = 0, a₃ = 0,

So they are linearly independent.

Example 6: Find the eigenvalues and orthonormal eigenvectors of A = 

For eigenvalues (A - λI) = 0

⇒ - (2 + λ) {-λ (1 - λ) -12} - 2 {-2λ - 6} - 3 {-4 + (1 - λ)) = 0
⇒ -(2 + λ){-λ + λ²-12} -2{-2λ-6}-3{-3-λ} = 0
⇒ (2 +λ) {-λ +λ² -12} - 2 {2λ + 6} - 3 {3 +λ} = 0
⇒ 2{-λ + λ²-12} + {-λ²+λ³-12λ)-7λ-21 = 0
⇒ λ³ +λ² -21λ-45 = 0 ⇒ (λ + 3)² (λ-5) = 0 ⇒ λ = -3, -3, 5

where λ₁ = 5 is non-degenerate eigenvalue and λ₂, = λ₃, = -3 is doubly degenerate eigenvalue.

Eigenvector can be determined by the equation AX = λX .


⇒ -2x₁ + 2x₂ - 3x₃ = 5x₁, 2x₁ + x₂ - 6x₃ = 5x₂ and x₁ + 2x₂ =-5x₃

⇒ -7x₁ + 2x₂ - 3x₃ = 0 - 2x₁ - 4x₂ - 6x₃ = 0 and x₁ + 2x₂ + 5x₃ = 0
From above relations x₂ = 2x₁ and x₃ = -x₁ ⇒ X₁ =
Normalized eigenvector can be determined by relation X₁^T X₁ = 1 ⇒
For λ₂ = λ₃ = -3
⇒ -2X₁ + 2x₂ - 3x₃ = -3X₁, 2X₁ + x₂ - 6x₃ = -3x₂ and -X₁ - 2x₂ = -3x₃
⇒ X₁ + 2x₂ - 3x₃ = 0, 2X₁ + 4x₂ - 6x₃ = 0 and x₁ + 2x₂ - 3x₃ = 0
Let x₃ = 0 ⇒ X₁ = -2x₂. So eigenvector is given by  for any value of x₂ .
Normalized eigenvector can be determined by relation X₂^TX₂ =1 ⇒ X₂ =
Now we must find third Eigenvector which will satisfy equation x₁ + 2x₂ - 3x₃ = 0 and orthogonal to X₂. X₂^TX₃ = 0 ⇒ [-2 1 0]
So X₃ = and value of x₁ can be find with orthogonal relation X₃^TX₃ = 1

The Cayley-Hamilton Theorem

This theorem provides an alternative method for finding the inverse of a matrix A . Also any positive integral power of A can be expressed, using this theorem, as a linear combination of those of lower degree.

Every square matrix satisfied its own characteristic equation. That means that, if

is the characteristic equation of a square matrix A of order n , then

Note:

When λ is replaced by A in the characteristic equation, then constant term an should be replaced by a_n to get the result of Cayley-Hamilton theorem, where I is the unit matrix of order n . Also 0 in the R.H.S is a null matrix of order n .
Example 7: Find A^-1 by Cayley-Hamilton theorem if A =

The characteristic equation of A is (A-λI) = 0

By Cayley-Hamilton theorem

Pre-multiplying by A^-1 we get A^-1 =

Example 8: Find A^-1 by Cayley-Hamilton theorem if A =

The characteristic equation of A is (A-λI) = 0

By Cayley-Hamilton theorem

Pre-multiplying by A^-1 we get A^-1 =

Example 9: Find A^-1 by Cayley-Hamilton theorem if A =

The characteristic equation of A is (A -λI) = 0

By Cayley-Hamilton theorem
A³ - A² - A + I = 0 ⇒ I = [A³ + A² + A]
Pre-multiplying by A^-1 we get A^-1 = [-A² + A + I]