Matrices and linear algebra

Question 1

What is the commutator?

Answer 1

[A,B]=AB-BA

Question 2

What does commutative mean?

Answer 2

AB=BA

Question 3

What three ways are there to change equations without changing the solution?

Answer 3

1. 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

Question 4

What are the aims for row-reduced echelon?

Answer 4

1. Want 1st non-zero entry in a row to be 1
2. Want the 1st non-zero entry for each row to be at he right of the previous row
3. Want each column that contains one of the leading 1’s to have all other entries zero

Question 5

What is the trace of a matrix?

Answer 5

Tr(A) equals the sum of the diagonal

Question 6

What are the properties of the trace?

Answer 6

Tr(A+B)=Tr(A)+Tr(B)
Tr(AB)=Tr(BA)

Question 7

What is the volume of a parallelepiped formed by 3 row vectors?

Answer 7

The determinant of the matrix containing those 3 row vectors?

Question 8

What determinant property allows to expand along the 1st column instead of 1st row?

Answer 8

det(A)=det(A’)
Where A’ is the transpose of A

Question 9

What determinant property allows to expand along any row or column?

Answer 9

If a matrix B is obtained by interchanging two rows or two columns of a matrix A then det(B)=-det(A)

Question 10

What is the determinant for a triangular matrix?

Answer 10

The product of all entries on the diagonal

Question 11

What is the co-factor rule?

Answer 11

det(A)=∑a_ij*c_ij
From j=1

Question 12

What is the false co-factor rule?

Answer 12

det(A)=∑a_ij*c_kj=0
From j=1 for all i ≠ k

Question 13

If two rows of a matrix, A, are the same what is det(A)?

Answer 13

det(A)=0

Question 14

What is the adjugate matrix?

Answer 14

Adj(A) equals the transpose of the co-factor matrix

Question 15

What is a singular matrix?

Answer 15

A matrix for which det(A)=0 and has no inverse

Question 16

What is the inverse of a matrix?

Answer 16

A^-1=Adj(A)1/det(A)
AAdj(A)=det(A)*I

Question 17

What are the properties of matrix inevrses?

Answer 17

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

Question 18

What is the LU decomposition method?

Answer 18

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)

Question 19

How do we use the LU decomposition method to solve simultaneous equations?

Answer 19

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

Question 20

What is a symmetric matrix?

Answer 20

When A equals the transpose of A

Question 21

What is an anti-symmetric matrix?

Answer 21

When the transpose of A equals -A

Question 22

What is an orthogonal matrix?

Answer 22

When the transpose of A equals the inverse of A

Question 23

What is the Hermitian conjugate matrix?

Answer 23

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

Question 24

What is a Hermitian matrix?

Answer 24

When A dagger equals A

Question 25

What is an anti-Hermitian amtrix?

Answer 25

When A dagger equals -A

Question 26

How are eigenvalues calculated?

Answer 26

det(A-Iλ)=0
The obtain a quadratic, cubic etc in λ and solve for λ

Question 27

How are eigenvectors calculated?

Answer 27

Ax=λ_i x
Equate these two and solve for a,b,c etc and then normalise the eigenvector

Question 28

What is orthogonality for vectors?

Answer 28

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

Question 29

What is a unitary matrix?

Answer 29

U dagger equals the inverse of U
The magnitude of λ_i squared =1

Question 30

What is the similarity transformation?

Answer 30

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