But! I can make this productive! I have a linear algebra quiz in about two and a half hours, and I can take notes here while I blog (and/or mindlessly type).
This is inspired by Yuma's rather nonfunctional blog now, and I would link, but it contains his real name and mine, which would make all of these nicknames rather silly and pointless.
Anyway.
Determinants and eigenvalues. The chapter where I actually learned new things.
- The most generic way of finding the determinant of a matrix is to take one row/column and add all the (entry * cofactor of entry) for that row/column.
- Cofactor = (-1)^(row# + column# of entry) det (matrix without entry's row and column)
- This technically works with 2 x 2 matrices, but what you're essentially doing is ad - cb anyway.
- This process is officially called the cofactor expansion. I always think of it as the "reduce the matrix the determinant way."
- SPECIAL CASES:
- if one row/column is all 0, then obviously expand along that row/column = det(matrix) = 0
- if the matrix is an upper/lower triangular matrix, multiply entries along diagonal (this one's easy to see)
- det(k*matrix) = k^n * det(matrix)
- det(transpose of matrix) = det(matrix) since rows/columns preserve
- det(matrix A * matrix B) = det(A) * det(B); and the extrapolation of det(matrix^k) = det(matrix)^k
- Row operations & determinants:
- 1. interchanging = -det(matrix)
- 2. multiplying one row by k = k det(matrix)
- 3. adding mults of one row to another = det(matrix)
- 1 because of signs of cofactors; 2 because of factoring; 3 because of same entries in two rows
- If det(A) = 0, no inverse (gasp!) and otherwise, yes inverse where det(A^-1) = 1/det(A) (sort of like the "inverse" of a number).
- Matrices can be blocked off to make calculations easier (if applicable). Only helps if you can make a square block of 0s.
- Adjoint of a matrix is the transpose of the cofactors matrix (matrix with cofactor of each entry in place of the corresponding entry).
- For matrix A: A * adj(A) = det(A) * I = adj(A) * A and adj(A) / det(A) = A^-1, where the proof of the second comes from the dividing everything by det(A).
- det(adj(A)) = detA^(n-1) where A is n x n; derived from above formula
- Cramer's Rule: so confusing to write... (to find nth x, replace nth column with constant column in coefficient matrix then find det (that) / det(A)) however it's more difficult to calculate for larger matrices and therefore only useful in theory
- Diagonalization is useful for calculating powers of square matrices (that can be diagonalized).
- If a square matrix can be diagonalized, then P^-1 * A * P = D, where P is an invertible "diagonalizing" matrix (so many terms) and D is the "diagonalized" matrix.
- P = eigenvectors of A and D = eigenvalues of A, in respective order (eigenvalue 1 matches with eigenvector 1 in column location).
- Eigenvalues: AX = xX where x is a constant (the eigenvalue) that is usually lambda but I can't type it easily.
- Generally, eigenvalues are found by solving the characteristic polynomial, which is det(xI - A) = 0.
- The eigenvectors are found by solving for X when the eigenvalues are plugged in for (xI - A) * X = 0.
- side note: tr(A) is the trace of A and is the sum of all diagonal entries
- For P to exist, # of eigenvectors = # of columns in A (or the total multiplicity of the eigenvalues, or the # of rows in P)
- Similar matrices are similar (no pun intended) to A and D pairs, but the other matrix need not be D.
- if A ~ B (~ = is similar to): A^-1, A^T, A^k ~ B^-1, B^T, B^k for k >= 0 respectively
- if A ~ B: det(A), c(x) of A, and eigenvalues of A = those of B; by this the matrix A has similar properties to the eigenvalues of A, e.g. if x^2 (the eigenvalue of A) = 5x, then A^2 = 5A.
- if A ~ B and B ~ C then A ~ C
My other work consists of writing the two paragraphs I have blocked out for my paper today (I have the topic sentences and the research, so that should be easy), as well as another location-specific paragraph that I need to do a bit more research on that hopefully won't take too long, and hopefully I can tackle another part of my paper (bring it to three full pages) as well.
Then I have reviewing for my Java midterm, so I need to start on my assignment as well, since it's been said that having that done before the midterm is good review.
My mom suggested I listen to French radio, so I could do that in my spare time. I also need to start researching for Robot On Moon (hey, ROM is also Royal Ontario Museum!), and sign up for a project for FSAE, and talk with the person in charge of notes, and review at least calc and hopefully physics too, and finish my letter and send out the food, and write my story, and fit in all my Halloween parties, and figure out my linear algebra assignment eventually, and stop typing "assignmnet" every time I mean "assignment" because terminal doesn't recognize it and spits at me every time I do so.
But. First. Fooooood.
