Overview: Matrix

Table of Contents
1. Introduction
2. Important formulas for matrices
3. Types of matrices
4. Trace of a matrix
5. Matrix transpose
6. Properties of matrix multiplication
7. Adjoint of a matrix
8. Inverse of a matrix
9. Order of a matrix and basic notes
10. Illustration 1: Construct a 3×4 matrix A = [aij], whose elements are given by aij = 2i + 3j.
11. Illustration 2: Construct a 3×4 matrix whose elements are given by: aij =
12. Trace - further properties and example
13. Transpose of a matrix - definition and example
14. Properties of transpose (summary)
15. Problems on matrices
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Introduction

An m × n matrix is a rectangular array of numbers with m rows and n columns. A typical m × n matrix A is written as

The matrix above may be represented compactly as A = [a_ij]_m×n. Each entry a_ij is called the (i, j)th element of A. The element a_ij lies in the i^th row and j^th column of the matrix.

Important formulas for matrices

Let A and B be square matrices of order n, and let I_n denote the n × n identity (unit) matrix. The following formulas and properties involving the adjoint (or classical adjoint) and determinant are useful.

(a) A(adj A) = |A| I_n = (adj A)A
(b) |adj A| = |A|^n-1
    (Thus A · adj A is always a scalar matrix.)
(c) adj(adj A) = |A|^(n-2) A
(d) adj(A + B) does not have a simple general expansion analogous to determinants; treat on a case-by-case basis.
(e) adj(AB) = (adj B)(adj A)
(f) adj(A^m) = (adj A)^m, for integer m ≥ 1
(g) adj(kA) = k^n-1adj A, for scalar k ∈ ℝ
(h) adj(I_n) = I_n
(i) adj(0) = 0
(j) If A is symmetric then adj A is also symmetric
(k) If A is diagonal then adj A is diagonal
(l) If A is triangular then adj A is triangular
(m) If A is singular then |adj A| = 0

Types of matrices

• Symmetric matrix: A square matrix A = [a_ij] is symmetric if a_ij = a_ji for all i, j.
• Skew-symmetric matrix: A square matrix A is skew-symmetric if a_ij = -a_ji for all i, j. In particular, diagonal entries of a real skew-symmetric matrix are zero.
• Hermitian and skew-Hermitian matrices: For complex matrices, a matrix A is Hermitian if A equals its conjugate transpose. The conjugate transpose is denoted in the original text as A^θ (that is, take the transpose and then the complex conjugate of each entry). A matrix is skew-Hermitian if it equals the negative of its conjugate transpose.
• 
  
• Orthogonal matrix: A square real matrix A is orthogonal if \(AA^{T}=I_n\) (equivalently \(A^{T}A=I_n\)), so \(A^{-1}=A^{T}\).
• Idempotent matrix: A is idempotent if \(A^{2}=A\).
• Involutory matrix: A is involutory if \(A^{2}=I\) (equivalently \(A^{-1}=A\)).
• Nilpotent matrix: A square matrix A is nilpotent if \(A^{p}=0\) for some positive integer p.

Trace of a matrix

The trace of a square matrix is the sum of the elements on its main diagonal. For an n × n matrix A = [a_ij],

The trace is denoted by tr(A).

• (i) tr(λA) = λ·tr(A) for any scalar λ.
• (ii) tr(A + B) = tr(A) + tr(B) for n × n matrices A and B.
• (iii) tr(AB) = tr(BA) whenever the products AB and BA are defined (square matrices of the same order).

Matrix transpose

The transpose of a matrix A is the matrix obtained by interchanging its rows and columns. The transpose is denoted by A^T or A'. If A has order m × n then A^T has order n × m.

Properties of transpose

• (i) \((A^{T})^{T}=A\).
• (ii) \((A+B)^{T}=A^{T}+B^{T}\).
• (iii) \((AB)^{T}=B^{T}A^{T}\).
• (iv) \((kA)^{T}=kA^{T}\) for scalar k.
• (v) \((A_{1}A_{2}\cdots A_{n})^{T}=A_{n}^{T}\cdots A_{2}^{T}A_{1}^{T}\).
• (vi) \(I^{T}=I\).
• (vii) tr(A)=tr(A^T).

Properties of matrix multiplication

• (i) Matrix multiplication is generally non-commutative: AB ≠ BA in general.
• (ii) Matrix multiplication is associative: \((AB)C=A(BC)\) whenever sizes are compatible.
• (iii) Multiplication is distributive over addition: \(A(B+C)=AB+AC\) and \((A+B)C=AC+BC\).

Adjoint of a matrix

The adjoint (classical adjoint) of an n × n matrix A, denoted by adj A or \(\operatorname{adj}A\), is the transpose of the cofactor matrix of A. Each entry of adj A is the cofactor of the corresponding entry of A, placed in transposed position.

Inverse of a matrix

An n × n matrix A has an inverse A^-1 if and only if A is non-singular, i.e., \(|A|\ne 0\). When the inverse exists, it satisfies \(AA^{-1}=A^{-1}A=I_n\).

The inverse of A (when \(|A|\ne 0\)) can be written in terms of the adjoint:

\(A^{-1}=\dfrac{1}{|A|}\operatorname{adj}A\).

Order of a matrix and basic notes

The order of a matrix is given by the number of rows by number of columns, denoted m × n. For example:

The matrix shown above has order 2 × 3.

• Note (a): A matrix is an arrangement of quantities in rows and columns.
• Note (b): Elements of a matrix may be real or complex. A matrix all of whose elements are real is called a real matrix.
• Note (c): An m × n matrix has m·n elements.

Illustration 1: Construct a 3×4 matrix A = [a_ij], whose elements are given by a_ij = 2i + 3j.

Sol:

Compute the element in the i^th row and j^th column by substituting i and j in the formula a_ij=2i+3j.

a₁₁=2·1+3·1=5

a₁₂=2·1+3·2=8

a₁₃=2·1+3·3=11

a₁₄=2·1+3·4=14

a₂₁=2·2+3·1=7

a₂₂=2·2+3·2=10

a₂₃=2·2+3·3=13

a₂₄=2·2+3·4=16

a₃₁=2·3+3·1=9

a₃₂=2·3+3·2=12

a₃₃=2·3+3·3=15

a₃₄=2·3+3·4=18

Illustration 2: Construct a 3×4 matrix whose elements are given by: a_ij =

Sol:

The method is identical to Illustration 1: substitute respective row and column indices i and j into the given formula for a_ij to obtain each entry.

Hence the required matrix is:

Trace - further properties and example

Let A = [a_ij]_n×n and B = [b_ij]_n×n, and let λ be a scalar. Then

• tr(λA) = λ·tr(A)
• tr(A + B) = tr(A) + tr(B)
• tr(AB) = tr(BA)

Transpose of a matrix - definition and example

The transpose of a matrix A is obtained by turning its rows into columns (or columns into rows). If A is m × n then A^T is n × m. For example, the transpose of the following matrix is shown:

Properties of transpose (summary)

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^TA^T
• (kA)^T = kA^T
• (A₁A₂ ··· A_n)^T = A_n^T ··· A₂^TA₁^T
• I^T = I
• tr(A) = tr(A^T)

Problems on matrices

Illustration 3: If

then prove that (AB)T = BTAT.

Solution:

Obtain the product AB explicitly (using the usual rule: row of A times column of B) and then form the transpose of the resulting matrix to get (AB)^T.

Obtain B^T and A^T individually by transposing B and A respectively.

Multiply B^T and A^T in that order to get B^TA^T.

Compare the entries of (AB)^T and B^TA^T; they are equal, which proves (AB)^T=B^TA^T.

Illustration 4: If

Then what is (B')'A' equal to?

Sol:

Use the properties of transpose. Note that (B')' means the transpose of the transpose of B, that is (B^T)^T=B.

Therefore (B')'A' = BA^T.