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)
Diagonal elements of symmetric matrix
Real
How to identify matrix is skew symmetric matrix if it is given
Diagonal elements are zero
Lower triangular elements = -Upper triangular elements
Involuntary matrix
A² = I
or
A = A⁻¹
Idempotent matrix
A² = A
Trace(k(A))
k(Trace(A))
Trace(A+B)
Trace(A)+Trace(B)
Matrix multiplication is associative or not
What it means
Associative
A(BC) = (AB)C
(AB)ᵀ
BᵀAᵀ
All symmetric matrix are ___________ matrix
Hermitian
If any matrix is multiplied for numbers of time so that the result is null matrix is called ________________
Nilpotent matrix
A is skew-symmetric matrix, what it tells about Aⁿ
Aⁿ is symmetric if n is even
Aⁿ is skew-symmetric if n is odd
Diagonal elements of Hermitian matrix
if elements of diagonal are real, then there is no affect on diagonal elements
A matrix is multiplied by itself and you are getting identity matrix then
matrix and its inverse both are equal
and matrix is involuntary matrix
Diagonal elements of Skew symmetric matrix
0
Skew symmetric matrix
A = -Aᵀ or A + Aᵀ=0
If a matrix is multiplied by itself and you are getting same matrix then such matrix is _________________
Idempotent matrix
Trace(Aᵀ)
Trace(A)
(Aᵀ)ᵀ
A
(A+B)ᵀ
Aᵀ+Bᵀ
order of Aᵀ
nxm
Nilpotent matrix
Aᵏ = Onxn (Null matrix)
here k is index or degree
For Trace(A), which condition is required
The matrix should be square
If AB = A and BA = B then
A and B both are idempotent matrices
Conjugate matrix of real elements
No affect on matrix
Hint of Hermitian matrix
tranjugate = transpose + conjugate
(-A)ᵀ
-Aᵀ
Unitary matrix
AAθ = Aθ A = I
or
Aθ = A⁻¹
(here θ is superscript)
Singular matrix
|A| = 0
Non-singular matrix
|A| ≠ 0
Inverse of matrix
A⁻¹ = adjA / |A|, where |A| != 0
(A+B)²
(A+B)² = A² + AB + BA + B²
The minimum number of multiplications required to multiply Aₘₓₙ with Bₙₓₚ
The minimum number of multiplications required to multiply Aₘₓₙ with Bₙₓₚ is mnp
Matrices of same dimension means
same order
i.e. if you have A of order mxn then order of B will be same i.e, mxn
If A and D are similar matrices, what about their determinants
|A|=|D|
(AB)ⁿ
BⁿAⁿ
(AB)⁻¹
B⁻¹A⁻¹
AAᵀ matrix
AAᵀ matrix
symmetric matrix
note about product of two symmetric matrices
Product of 2 symmetric matrices is need no to be symmetric unless A = B
note about product of two skew symmetric matrices
Product of 2 skew-symmetric matrices is need no to be skew symmetric unless AB = -BA
Geometrical definition of orthogonal matrix
A = [x₁ x₂ x₃ … xₙ]ₙₓₙ is a orthogonal matrix whose vectors are orthonormalisation vectors.
[Here x₁,x₂,x₃, … xₙ are n tuple vectors i.e. they are n-dimensional column vectors i.e. have n components]
OR
A square matrix whose vectors are orthonormal vectors called orthogonal matrix.
Partition matrices
Divide the matrix into number of blocks
Type of partition matrices
also their determinants
and condition for it also
Inverse of matrix (Proper formula)
Inverterse of 2x2 matrix (shortcut)
