Important Formulas: Matrices

Matrices

A matrix is a rectangular arrangement of numbers in m rows and n columns, referred to as a matrix of order m × n.

In this matrix, a_ij represents the element in the i^th row and j^th column. This matrix is denoted as [a_ij]_m×n. The elements a₁₁, a₂₂, a₃₃, etc., are the diagonal elements. The sum of these diagonal elements is referred to as the trace of A, symbolized by Tr(A).

Fundamental Definitions

• Row matrix: A matrix with only one row.
• Column matrix: A matrix with only one column.
• Square matrix: A matrix of order m×n where m = n.
• Zero matrix: A = [a_ij]_m×n is a zero matrix if a_ij = 0 for all i and j.
• Upper triangular matrix: A = [a_ij]_m×n is an upper triangular matrix if a_ij = 0 for i > j.
• Lower triangular matrix: A = [a_ij]_m×n is a lower triangular matrix if a_ij = 0 for i < j.
• Diagonal matrix: A square matrix [a_ij]_m×n is a diagonal matrix if a_ij = 0 for i ≠ j.
• Scalar matrix: A diagonal matrix A = [a_ij]_m×n is a scalar matrix if a_ij = k for i = j.
• Unit matrix (Identity matrix): A diagonal matrix A = [a_ij]_n is a unit matrix if a_ij = 1 for i = j.
• Comparable matrices: Two matrices A and B are comparable if they have the same order.

Important Concepts and Operations

1. Equality of matrices: Two matrices A = [a_ij]_m×n and B = [b_ij]_p×q are equal if m = p and n = q and a_ij = b_ij for all i and j.

2. Scalar multiplication of a matrix: If λ is a scalar, then λA = [b_ij]_m×n where b_ij = λa_ij for all i and j.

3. Addition of matrices: If A = [a_ij]_m×n and B = [b_ij]_m×n are two matrices, then A+B = [a_ij]_m×n + [b_ij]_m×n = [c_ij]_m×n where c_ij = a_ij +b_ij for all i and j.

4. Subtraction of matrices: A-B = A+(-B), where -B = ( -1)B.

5. Properties of addition and scalar multiplication:

• λ(A+B) = λA+λB
• λA = Aλ
• (λ₁+λ₂)A = λ₁A+λ₂A

6. Multiplication of matrices: If A = [a_ij]_m×p and B = [b_ij]_p×n, then AB = [c_ij]_m×n where

7. Properties of matrix multiplication:

• AB ≠ BA
• (AB)C = A(BC)
• AI_n = A = I_nA
• For every non singular square matrix A (i.e., | A |≠ 0 ) there exists a unique matrix B such that AB = I_n = BA. We say that A and B are multiplicative inverses of each other. That is, B = A^-1 or A = B^-1.

8. Transpose of a Matrix.
If A = [a_ij]_m×n then the transpose of A, represented by A' or A^T, is defined as A' = [a_ji]_n×m.

• (A')' = A
• (λA)' = λA'
• (A+B)' = A'+B'
• (A-B)' = A'-B'
• (AB)' = A'B'
• For a square matrix A, if A' = A, then A is a symmetric matrix.
• For a square matrix A, if A' = -A, then A is a skew symmetric matrix.

9. Submatrix of a matrix: A submatrix is the matrix obtained by deleting certain rows and columns from a given matrix A.

10. Properties of determinant:

• | A | = | A' | for any square matrix A.
• If two rows or two columns are identical, then | A | = 0.
• | λA | = λⁿ| A |, when A = [a_ij]_n×n.
• If A and B are two square matrices of the same order, then | AB |= | A || B |

11. Singular and Non-singular matrix: A square matrix A is singular if | A | = 0 and non-singular if | A | ≠ 0.

12. Cofactor and adjoint matrix.
The cofactor matrix of a square matrix A = [a_ij]_n×n is obtained by replacing each element of A by its corresponding cofactor. The transpose of the cofactor matrix of A is called the adjoint of A, denoted as adj A.

13. Properties of adj A.

• A . adj A = | A |I_n = (adj A)A where A = [a_ij]_n×n.
•  | adj A | = | A |^n-1, where n is the order of A.
• If A is a symmetric matrix, adj A is also symmetric.
• If A is singular, adj A is also singular.
• For a non singular matrix A, the multiplicative inverse of A is given by
  
• (A^-1)^T = (A^T)^-1 for any non singular matrix.
• (A^-1)^-1 = A, if A is non singular.
• A^-1 is always non singular.
• (adj A^T) = (adj A)^T
• For a non zero scalar k and a non singular matrix A,
  
• |A^-1| = 1/|A| for | A | ≠ 0.
• For a non singular matrix A, if AB = AC, then B = C and if BA = CA, then B = C.

14. System of linear equations and matrices: A system of linear equations AX = B is consistent if it has at least one solution.
(i) System of linear equations and matrix inverse:

• If A is non-singular, the solution is given by X = A^-1B.
• If A is singular, adj(A) B = 0 and no two columns of A are proportional, the system has infinitely many solutions.
• If A is singular and (adj A)B ≠ 0, the system has no solution.

(ii) Homogeneous system and matrix inverse:

• If the above system is homogeneous (i.e., b₁ = b₂ =..... b_n = 0), then in the matrix form it is AX = 0, where A is a square matrix.
• If A is non-singular, the system has only one trivial solution, X = 0.
• If A is singular, then the system has infinitely many solutions, including the trivial solution, hence it has non-trivial solutions.

(iii) Elementary row transformation of Matrix:
Elementary row transformations of a matrix include the following operations:

• Interchanging two rows.
• Multiplying all the elements of a row by a non zero scalar.
• Adding a constant multiple of a row to another row.

15. Characteristic Polynomial and characteristic equation.
For a square matrix A, the polynomial | A-xI | is the characteristic polynomial of A and the equation | A-xI | = 0 is the characteristic equation of A.

16. Cayley Hamilton theorem:
Every square matrix A satisfies its characteristic equation. That is, if a₀xⁿ + a₁x^n-1 + ..... + a_n-1x + a_n = 0 is the characteristic equation of A, then a₀Aⁿ + a₁A^n-1 + ........+ a_n-1A + a_nI = 0.

17. More definitions on matrices:

• Nilpotent matrix: A square matrix A is nilpotent if A^p = 0 for some positive integer p. If p is the smallest such positive integer, then p is its nilpotency.
• Idempotent matrix: A square matrix A is idempotent if A² = A.
• Involutory matrix: A square matrix A is involutory if A² = I.
• Orthogonal matrix: A square matrix A is orthogonal if A^TA = I = AA^T.
• Unitary matrix: A square matrix A is unitary if A(Ā)^T = I, where Ā is the complex conjugate of A.