Matrix Flashcards
Trace(A)
sum of principal diagonal elements
Orthogonal matrix
A⁻¹ = Aᵀ
or AAᵀ = AᵀA = I
Matrix multiplication is commutative or not
What it means
Non commutative
AB != BA
Symmetric matrix
Aᵀ = A
(Aᵀ)ⁿ
(Aⁿ)ᵀ
If A is idempotent matrix then what it tells about Aⁿ
then Aⁿ = A
How to identify matrix is symmetric if it is given
The elements of upper triangular matrix and lower triangular matrix are symmetric about principal diagonal
Diagonal elements of matrix are real
What is index or degree k of Nilpotent matrix
How many times it is required to multiply to get the null matrix is understand as degree of it
Hermitian matrix
Ā = Aᵀ (A bar = A transpose)
or
(Ā)ᵀ = A
or
Aθ = A (A raise to the power theta = A)
A is symmetric, what it tells about Aⁿ
It is also symmetric
If AB = B and BA = A then
A and B both are idempotent matrices
Trace(A-B)
Trace(A)-Trace(B)
Conjugate of a matrix
it changes the sign of imaginary part of the complex element
How to represent conjugate matrix
Ā (A bar or A with macron)
Every square matrix can be written as
1.) _______________________
2.) _______________________
1.) Sum of symmetric and skew symmetric matrices
Anxn = 1/2(A + Aᵀ) +1/2(A-Aᵀ)
2.) Sum of hermitian and skew-hermitian matrix
Anxn = 1/2(A + Aθ) + 1/2(A - Aθ).
(here θ is in power)
complex matrix
A matrix having at least one element complex
Skew Hermitian matrix
Ā = -Aᵀ (A bar =- A transpose)
or
(Ā)ᵀ = -A
or
Aθ = -A (A raise to the power theta = -A)
