after midterm 2 material

Question 1

what are the steps to determine if the vectors are eigen vectors

Answer 1

• multiply the matrix by the vector (dot prod) form a vector
• make that new vector equal the multiplied vector with a variable coefficent
• make a equation for each element of the vector and solve for the varaible
• if the answers for the variables is not the same then it is not a eigen vector

Question 2

how to find the eigenvlaues of a matrix

Answer 2

• first subtract the eigenvariable from both the top left and bottom right values of the matrix
• then find the determinate of the matrix
• factor it out or use quadratic to find the eigen values

Question 3

how to find the eigenvector with the eigen values

Answer 3

• pull up the original matrix that is has the eigenvariable in it
• add the eigen vlaue into the matrix only using one eigen value per matrix
• solve and then take the dot product of the matrix and vector [x,y] and make that equal zero
• you get two equations solve for x and y
• put the x and y values in a vector and thats the eigen vector

Question 4

what is the dominate eigen value in a matrix

Answer 4

the dominate eigen vlaue is the largest one in the matrix on a absolute basis

Question 5

how to take the determinate of a 3x3 triangular matrix

Answer 5

just multiply the the diagonal values

Question 6

how to find the cofactor matrix

Answer 6

swap the place of A & D and make b & c negative

Question 7

how to check if a matrix is a markov matrix

Answer 7

• all entries are non negative
• the sum of each column is one

Question 8

how to find the fixed state of a markov matrix

Answer 8

• subtract the top left and bottom right values by one
• multiply the matrix by 10 to get whole numbers
• add the two positive values that number will be the denominator later
• form a vector with the top right value divided by the addition number as the top value and the bottom value as the bottom value in the vector

Question 9

how to determine if a set of vectors is a orthogonal set

Answer 9

• check if the vectors are orthogonal using the dot product nethod

Question 10

how to normalize a vector and find the orthanormal set

Answer 10

• do the following for all the values in the vector
• divide the element of the vector by the norm of the whole vector
• input those fractions into a vector and a set of those vectors is now the orthanormal set

Question 11

how to check if 3 vectors (A,B,C) are orthogonal

Answer 11

take the dot product of (A,B) (A,C), (B,C)

Question 12

how to know if a matrix is a probablity matrix

Answer 12

all the elements equal 1
the elements are non negative

Question 13

what is the dimension

Answer 13

its the number of vectors