Matrices Flashcards
Theorem: Diagonal matrices in terms of subscripts
Aij = aiδij
Definition: Matrix Addition
(A + B)ij := Aij + Bij
Theorem: Matrix addition is commutative
A + B = B + A
Theorem: Matrix addition is associative
A + (B + C) = (A + B) + C
Definition: Matrix Multiplication
(AB)ij =
(m) Sum(k=1)
AikBkj
Theorem: Properties of Matrix Multiplication
(i) A(BC) = (AB)C
(ii) A(B + C) = (AB) + (AC)
(iii) (A + B)C = AC + BC
Definition: Scalar Matrix
Aij = aδij
Theorem: Scalar Matrix using the identity
A = λI^(n)
Theorem: Diagonal matrices commute
AB = BA
Diagonal
Theorem: Scalar matrices commute with any other matrix
AB = BA
Scalar
Definition: Scalar Multiplication
(λA)ij = λAij
Theorem: Properties of scalar multiplication
(i) 1A = A
(ii) 0A = 0^(n,m)
(iii) λ(µA) = (λµ)A
(iv) (λA)(µB) = (λµ)AB
(v) λ(A + B) = λA + λB
(vi) (λ + µ)A = λA + µA
Definition: Transpose of a matrix
(A^T)ij := Aji
Theorem: Properties of the transpose
(i) (A^T)^T = A
(ii) (AB)^T = B^TA^T
(iii) (λA + µB)^T = λA^T + µB^T
Definition: Symmetric Matrix
A = A^T
Also square
Theorem: Diagonal matrices are always symmetric
A diagonal matrix =⇒ A is symmetric
Definition: Inverse of a matrix
AB = I^(n) = BA
Theorem: Determinant and Invertibility
A is invertible ⇐⇒ det(A) != 0
Theorem: Properties of the inverse
(i) (A^−1)^-1 = A
(ii)) (AB)^−1 = B^−1A^-1
(iii) (AT)^−1 = (A^−1)^T
(iv) (λA)^−1 =1/λ(A^−1)
(v) Assume AB = I^(n) ==> Then BA = I^(n)
Definition: Orthogonal Matrix
A^−1 = A^T
Special Matrix
