Flashcards
Basics
Cards in this deck (30)
- 1Conjugate TransposeAll elements of a matrix are complex conjugated, and the matrix is transposed
- 2Transpose, Conjugate Transpose
- 3Involutory Idempotent NilpotentA^2=I
A^2=A
A^k=0, k>0 - 4Inner product Outer productx*y=scalar
xy*=matrix (timetable for x and conj(y)) - 5Orthogonal vectorsx*y=0 (Inner product of 0)
- 6Orthonormal vectorsy*x=0
y*y=x*x=1 - 7Length of a vectorsqrt(x*x)
- 8Diagonal MatrixSquare matrix with only non-zero elements in diagonal
- 9Upper triangular Lower triangularAll elements below diagonal is zero
All elements above diagonal is zero - 10Block matrixPartioning of matrix into sub matricies
- 11SubmatrixA matrix where some of the columns/rows are deleted
- 12Block Diagonal Block triangular
- 13Symmetric MatrixA is square, A^T=A
- 14Hermitian matrixA is square, A*=A
- 15Positive definite Positive semi-definite indefinitex*Ax > 0 forall x
x*Ax >= 0 forall x
(x*Ax)(y*Ay) < 0 for some x,y (Both positive and negative numbers can accur) - 16Kronecker deltadelta_{i,j} = 1 if i==j else 0
- 17Orthogonal MatrixQ is real and square
QQ^T=Q^TQ=I
columns of Q are orthonormal
Q1*Q2 is also orthogonal - 18Unitary MatrixU is complex and square
U*U=UU*=I - 19Permutation Matrix
- 20Rank(A)Maximum number of linear independent rows or columns of A
or
dim(range(A)) - 21range(A){y in C: y=Ax, for some x},
ie. the span of the columns - 22null(A){x in C: Ax=0}, null space of A
- 23n - rank(A)?dim(null(A))
- 24det(AB) det(A+B) det(A^T) det(alpha*A) det(Q) det(U)det(A)det(B)
det(A+B)
det(A)
alpha^n*det(A)
+/-1
On the unit circle - 25det(A), A is block diagonal/triangularproduct of determinants of diagonal blocks
- 26InverseB is inverse of A (A^{-1}) if AB=I
- 27Singular non-singularA is non-singular if a inverse exists; otherwise singular
- 28What is equivelent of A being non-singular?null(A) = {0}, (Ax = 0 iff x=0)
rank(A) = n (rows or columns of A are linearly independent)
det(A) =/= 0
no eigenvalues are 0 - 29trace(A)sum of its diagonal elements
- 30trace(AB) trace(alpha*A+beta*B)trace(BA)
alpha*trace(A)+beta*(trace(B))
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- Matrix Analysis (Basics) Flashcards on Basics, created by Simon Majgaard on 07/11/2020.
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- Simon Majgaard
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