Eigenvalues and Vectors Flashcards
for a non-zero vector x that obeys Ax = Yx, where A is a matrix and Y is a constant lambda, what are x and Y called in this case
- x is the eigenvector of A
- Y is the corresponding eigenvalue
what is an eigenvalue
a scalar that determines the amount its corresponding eigenvector gets stretched / squashed
can eigenvectors and values = 0
- eigenvectors cannot = 0
- eigenvalues can = 0
if the e-value determines whether the e-vector gets stretched or squashed, what is the only thing about the e-vector that matters
its direction
what is usually the first step in e-vector and value problems
finding the e-values
if youre starting with Ax = Yx, what do you need to rearrange it to in order to start getting somewhere
- x(A - YI) = 0
- I is the identity matrix so all it does it give the e-values a matrix form
for an nxn matrix, how many different solutions for Y are there
n different solutions
when you have your A = YI matrix, how do you solve for Y
- the determinant of the matrix is 0
- so equate the equation for the determinant to 0 and slove
after you have found the e-values, what is the next step
finding the e-vectors that correspond to its e-value
you have now inputted one of the e-values into the x(A-YI) = 0 formula, however the det of the matrix = 0 so we cant just multiply out and solve simultaneously. in the case of a 2x2, what do we do next
- you remove the bottom row of the A-YI matrix and the (0,0) matrix at the end
- then you equate the bottom x value (x2) to 1
- then you expand and solve
for any nxn A-YI matrix with n length x vector, what is the general rule for doing this
- set x(n) to 1 in the x vector
- take out the last column from the nxn matrix and put it on the RHS of the equation
- multiply that column by -1
- remove the dangling 1 you set x(n) to completely
after you have done this again with the other e-value ( or e-values), what should you be left with
- an nx1 e-vector corresponding to 1 e-value
- with an n number of these pairs
what is the final step to solving problems like these
normalise the e-vectors AKA turn them into unit vectors
what does it mean if a matrix is symmetric
- it equals its transpose
- S = S^T
what does it mean if a matrix is anti-symmetric
A^T = -A
what is a defective (jordan) matrix
- an nxn matrix that has n e-values but not n e-vectors
- this happens when a matrix has repeated e-values
- in these cases the number of e-vectors is less than the number of e-values
let A be a 3x3 matrix with e-values Y1 Y2 and Y3 corresponding with e-vectors u1 u2 u3. if you were to combine the e-vectors into one nxn matrix by sticking them next to each other [u1,u2,u3], what would be the relationship between that matrix, A and the e-values
- A*[u1, u2, u3] = [Y1u1, Y2u2, Y3u3]
- u1 u2 and u3 are vertical 3x1 vectors
a matrix V (actually meant to be a capital lambda) is formed by making the diagonals the e-values, and all the other values in the matrix are 0. if the matrix of the combined e-vectors is U, how can the previous formula be simplified
AU = UV
what condition needs to be met in order for both sides to be post-multiplied by U^-1
the inverse of U must exist
considering the condition of Q^-1 = Q^T in this case as U is an orthogonal matrix, what would be the formula when multiplied by U^-1
A = UVU^T
what would A’ equal
A’ = V
what does V actually represent in this case
- it represents the same physical transformation as A
- but in a coordinate system aligned with e-vectors
what is the main feature of diagonal matrices when used as transformations
they correspond to pure stretching and compressing along axes
why cant defective matrices be diagonalised
because they dont have a full set of linearly independent vectors
what is special about the determinant of A in the formula
- its determinant is the product of its e-values
- which determines the scale factor of the mapping
going back to the form A = UVU^-1, what would A^n equal
A^n = UV^nU^-1
what would V^n look like
youd just raise the diagonal e-values to the nth power
what is a non-symmetric matrix
a matrix that tends to have complex e-values and vectors
what is the formula for getting started on finding the e-values of non-symmetric matrices
det(A-YI) = 0
how would you then find the e-vectors
- x(R-YI) = 0
- where R-YI is just the discovered values of Y put into the matrix
a non-zero vector x that satisfies Ax = YMx is an e-vector of the above generalised problem, and Y is the corresponding e-value. if A and M are symmetric, what does [(xi))^T * A * (xj)]^T equal
- [(xi))^T * A * (xj)]^T = (xj)^T * A^T * xi
- the order of the variables matter
