Matrices and linear algebra Flashcards
What is the commutator?
[A,B]=AB-BA
What does commutative mean?
AB=BA
What three ways are there to change equations without changing the solution?
- Interchange two equations/rows
2.Replace a line/row by itself +C times another equation/row
3.Multiply any equation/row by a non-zero constant
What are the aims for row-reduced echelon?
- Want 1st non-zero entry in a row to be 1
- Want the 1st non-zero entry for each row to be at he right of the previous row
- Want each column that contains one of the leading 1’s to have all other entries zero
What is the trace of a matrix?
Tr(A) equals the sum of the diagonal
What are the properties of the trace?
Tr(A+B)=Tr(A)+Tr(B)
Tr(AB)=Tr(BA)
What is the volume of a parallelepiped formed by 3 row vectors?
The determinant of the matrix containing those 3 row vectors?
What determinant property allows to expand along the 1st column instead of 1st row?
det(A)=det(A’)
Where A’ is the transpose of A
What determinant property allows to expand along any row or column?
If a matrix B is obtained by interchanging two rows or two columns of a matrix A then det(B)=-det(A)
What is the determinant for a triangular matrix?
The product of all entries on the diagonal
What is the co-factor rule?
det(A)=∑a_ij*c_ij
From j=1
What is the false co-factor rule?
det(A)=∑a_ij*c_kj=0
From j=1 for all i ≠ k
If two rows of a matrix, A, are the same what is det(A)?
det(A)=0
What is the adjugate matrix?
Adj(A) equals the transpose of the co-factor matrix
What is a singular matrix?
A matrix for which det(A)=0 and has no inverse
What is the inverse of a matrix?
A^-1=Adj(A)1/det(A)
AAdj(A)=det(A)*I
What are the properties of matrix inevrses?
If A and B are invertible then (AB)^-1=B^-1 * A^-1
If A is invertible then the transpose of A is also invertible
What is the LU decomposition method?
A matrix A is split into a lower matrix L and an upper matrix U
The the det(A)=det(L)*det(U)=det(U)
How do we use the LU decomposition method to solve simultaneous equations?
Ax=b
LUx=b
Ux=y
Ly=b
Use L and vector b to work out vector y
Then use vector y and U to work out vector x
What is a symmetric matrix?
When A equals the transpose of A
What is an anti-symmetric matrix?
When the transpose of A equals -A
What is an orthogonal matrix?
When the transpose of A equals the inverse of A
What is the Hermitian conjugate matrix?
The Hermitian conjugate of A is A dagger which equals the transpose of the complex conjugate of A as well as the complex conjugate of the transpose of A
What is a Hermitian matrix?
When A dagger equals A
What is an anti-Hermitian amtrix?
When A dagger equals -A
How are eigenvalues calculated?
det(A-Iλ)=0
The obtain a quadratic, cubic etc in λ and solve for λ
How are eigenvectors calculated?
Ax=λ_i x
Equate these two and solve for a,b,c etc and then normalise the eigenvector
What is orthogonality for vectors?
For a vector y the transpose of y dotted with y=1 and the transpose of y dotted with a vector x=0
For complex vectors replace the transpose with the Hermitian conjugate
What is a unitary matrix?
U dagger equals the inverse of U
The magnitude of λ_i squared =1
What is the similarity transformation?
A goes to S^-1 A S
Where S is a matrix made up of the eigenvectors of A
Which diagonalises a matrix A
If A is Hermitian then S^-1=S dagger
