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20FXandersguidetoeverythingpastchapter4 - The Chapter...

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The Chapter 5.1-5.3 Super Summary Chapter 5.1. The eigenvectors of a matrix are the non-zero vectors such that is some scalar multiple of . If is an eigenvector, then its eigenvalue is the scalar where . The eigenvalues of a matrix is the set of possible eigenvalues for all its eigenvectors. For any matrix and any eigenvalue of , the eigenspace of corresponding to is the set of all eigenvectors of with eigenvalue . All eigenspaces are vector spaces. If is triangular (either upper- or lower-), its eigenvalues are the entries on its main diagonal. This counts for diagonal matrices as well. Chapter 5.2. There’s a new way to find the determinant of a square matrix. Use row reductions to put it in row echelon form, except: Do NOT multiply a row in-place. (You can still add a multiple of a row to another) Keep track of how many times you exchange rows. Once you finish this, multiply the diagonal terms together (if you end up with a zero in a diagonal, your matrix is not invertible and has determinant zero). Then, multiply by , where is the number of rows exchanges you used. The result will be your determinant. Alternatively, you could multiply rows in-place, so long as you keep track of the multiples. For example, if you divide a row by 2 while row-reducing, you should multiply your final result by 2 to get the determinant. If you divide a row by 3 and multiply another by 4, you’ll have to multiply your result by 3 then divide by 4. Remember, you only need to keep track of this if you multiply a row in-place ; adding a multiple of a row to another can be done freely. Also, never multiply a row by zero -- that would mean you’d have to divide your end result by it! The following are equivalent (if one is true, they all are true): The matrix is not invertible. The number is an eigenvalue of Recall that:
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The characteristic equation of is: is an eigenvalue of if and only if it’s a solution of the characteristic equation. The number of times a factor appears in the characteristic equation is the multiplicity of the eigenvalue. For example, if this is your characteristic equation: Then the eigenvalue has multiplicity 4, the eigenvalue has multiplicity 2, and the eigenvalue has multiplicity 1. The dimension of an eigenspace is, at the maximum , the multiplicity of the eigenvalue. It could be anything between this value and 1. For any matrices and , if there exists an invertible matrix such that: Then and are similar (some call them conjugate ). Similar matrices have the same characteristic equation, and the same eigenvalues (but not the same eigenvectors!) Chapter 5.3. Let be an matrix. is called diagonalizable if it’s similar to a diagonal matrix, i.e. if there exists an invertible matrix and an diagonal matrix , such that: The following are all equivalent, for any matrix : is diagonalizable.
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20FXandersguidetoeverythingpastchapter4 - The Chapter...

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