Unit 2 - Inverting Matrices Flashcards
Dyadic product
A dyadic product of two vectors is a third vector product next to dot product and cross product. The dyadic product is a square matrix that represents a tensor with respect to the same system of axes as to which the components of the vectors are defined that constitute the dyadic product
When can be matrix A and B multiplied?
A type m x p
B type p x n
When the number of columns of the first matrix is the same as the number of rows od the second matrix.
We obtain an m x n matrix.
Diagonal matrices scaling
- If A is multiplied from the left by a diagonal matrix D, the rows are scaled with the elements of the diagonal.
- If A is multiplied from the right by a diagonal matrix D, the columns are scaled with the elements of the diagonal.
Matrix product transposed (AxB)^T
(A*B)^T = B^T * A^T
Calculation rules for transposition (5)
(A^T)^T = A
(A+B)^T = A^T + B^T
(λA)^T = λA^T
(A*B)^T = B^T * A^T
E^T = E
Identity matrix E
A square neutral matrix whose product with any matrix yelds the same matrix again.
Is a diagonal matrix whose main diagonal elements are 1.
Matrix multiplication law: Associative law
(AB)C=A(BC), if the size of the matrices permit it.
Matrix multiplication law: Homogenity
λ(AB) = (λA)B = A(λ*B)
Matrix multiplication law: Distributive law
Left hand: A(B+C)=AB+AC
Right hand (A+B)C=AC+BC
Matrix multiplication law: Square identity
E*A=A
Matrix multiplication law: Non-Square identity
A*E=A
Matrix multiplication law: Zero matric product
0A=0
A0=0
Matrix multiplication law: Scalar multiplication
λA=(λE)*A
How is defined a power of a square matrix?
It’s defined like the power of a normal number, as repeated multiplication.
Power properties
A^0=E, A^1=A, A^2=AA
A^s+t = A^s * A^t
A^st = (A^s)^t
