Matrix Operations: Scalars, Sums, and Multiplication

Matrix Operations

• If matrix A is m × n, the scalar at row i and column j of A, denoted aᵢⱼ, is the (i, j) coefficient of A.
• Each column of A is a list of m real numbers, defining a vector.
• Columns are denoted a₁, ..., aₙ, where matrix A can be represented as A=[a₁ a₂ ... aₙ].
• The real number aᵢⱼ is the i-th coefficient (from the top) of the j-th column vector aⱼ.
• Diagonal coefficients of an m × n matrix A = [aᵢⱼ] are a₁₁, a₂₂, a₃₃, ..., forming the main diagonal of A.
• A diagonal matrix is an n × n matrix with all non-diagonal coefficients being zero.
  • The identity matrix Iₙ is an example (Iᵢᵢ = 1).

Sum and Scalar Multiplication

• A m × n matrix with all elements as zero is the zero matrix, denoted 0.
• Two matrices A and B are equal if they have the same dimension (same number of rows and columns) with aᵢⱼ = bᵢⱼ.
• For m × n matrices A and B, the sum A + B is a matrix of dimension m × n with columns as the sums of the corresponding columns of A and B.
• The coefficients of A + B are the sums of corresponding coefficients of A and B.
• The sum A + B is defined only if A and B have the same dimension (same size).
• If r is a scalar and A is a matrix, the product by a scalar rA is the matrix with columns obtained by multiplying corresponding columns of A by r.
• For matrices A, B, and C of the same dimension, and scalars r and s following theorems apply;
  • A + B = B + A
  • (A + B) + C = A + (B + C)
  • A + 0 = A
  • r(A + B) = rA + rB
  • (r + s)A = rA + sA
  • r(sA) = (rs)A
• Each equality can be verified by showing the left and right sides have the same dimension and corresponding columns are equal.

Matrix Multiplication

• Multiplication by matrix B transforms vector x into vector Bx.
• Vector x multiplied by matrix B then matrix A is A(Bx).
• A(Bx) is obtained from x through a composition of linear applications.
• The application is represented by a single matrix AB, where A(Bx) = (AB)x.
• If matrix A is m × n, B is n × p, and x is in ℝp, the columns of B are denoted b₁, ..., bp and the components of x are x₁, ..., xp.
  • Bx = x₁b₁ + ... + xpbp.
  • A(Bx) = A(x₁b₁) + ... + A(xpbp) = x₁Ab₁ + ... + xpAbp.
• Vector A(Bx) is a linear combination of vectors Ab₁, ..., Abp, with coefficients as the components of x.
  • In matrix notation, this combination is written as A(Bx) = [Ab₁ Ab₂ ... Abp]x.
• Multiplication by [Ab₁ Ab₂ ... Abp] transforms x into A(Bx).

Definition of Matrix Multiplication

• If A is an m × n matrix and B is an n × p matrix with columns b₁, ..., bp, the product AB is an m × p matrix with columns Ab₁, ..., Abp.
  • AB = A[b₁ b₂ ... bp] = [Ab₁ Ab₂ ... Abp]
• Matrix multiplication corresponds to the composition of linear applications.
• Each column of AB is a linear combination of columns of A with coefficients as the components of corresponding columns of B.

Row-Column Rule for AB Calculation

• If the product AB is defined, the coefficient of the i-th row and j-th column of AB is the sum of the products of components of the i-th row of A and j-th column of B.
  • If (AB)ᵢⱼ is the coefficient (i, j) of AB, and A is an m × n matrix:
    • (AB)ᵢⱼ = aᵢ₁b₁ⱼ + ... + aᵢₙbₙⱼ

Proprieties of Matrix Multiplication

• Given matrix A is m × n, and B and C are such that the product and sum are defined:
  • A(BC) = (AB)C (associativity of multiplication)
  • A(B+C) = AB + AC (left distributivity)
  • (B+C)A = BA + CA (right distributivity)
  • r(AB) = (rA)B = A(rB) (for any scalar r)
  • IₘA = A = AIₙ (neutral element (unit) for multiplication)
• Associativity follows from matrix multiplication corresponding to composition of linear applications/functions.
• If C = [c₁ ... cp], then BC = [Bc₁ ... Bcp] and A(BC) = [A(Bc₁) ... A(Bcp)].

Additional Matrix Properties

• A(Bx) = (AB)x, then A(BC) = [(AB)c₁ ... (AB)cp] = (AB)C.
• Order is important as AB and BA are generally different.
• Columns of AB are linear combinations of columns of A, while BA's columns are linear combinations of columns of B.
• In AB, A is multiplied on the right by B, or B is multiplied on the left by A.
• If AB = BA, A and B commute.

Important Notes on Matrix Multiplication

• AB ≠ BA in general.
• Matrix simplification is limited; AB = AC does not imply B = C.
• AB equaling a zero matrix does not imply A = 0 or B = 0.

Matrix Powers

• If A is an n × n matrix and k is a strictly positive integer, Ak is the product of k matrices equal to A: Ak = A...A.
• If A is a non-zero matrix and x is in ℝⁿ, Akx is obtained by multiplying x, k times on the left by A.
• If k = 0, A⁰x is x itself.
  • A⁰ is the identity matrix (unit).

Matrix Transpose

• In a m × n matrix, the transpose of A is the n × m matrix Aᵀ, obtained by exchanging the columns and rows with the same index i of A.
• The transpose of the product of two matrices equals the product of their transposes in reverse order.

Theorem Relating A and B Transposes

• Given matrices A and B with compatible dimensions:
  • (Aᵀ)ᵀ = A
  • (A + B)ᵀ = Aᵀ + Bᵀ
  • For any scalar r, (rA)ᵀ = rAᵀ
  • (AB)ᵀ = BᵀAᵀ