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36 Cards in this Set
- Front
- Back
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What are 7 properties of invertible matrices?
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1. They are square
2. det is not equal to 0, eigenvalues, not equal to 0 3. All orthogonal matrices are invertible 4. columns for a basis of R^n 5. A^T•A is invertible 6. Invertible matrices reduce to identity matrix 7. A and A^-1 have the same eigenvectors |
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If 2 matrices are invertible what form of A has the same eigenvectors as A?
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A^-1
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If A is invertible, is A^3 invertible?
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Yes
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If invertible A^2 = A, what does A equal?
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A=In
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Do similar matrices have the same eigenvalues?
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Yes
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Do simliar matrices have the same algebraic multiplicities?
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Yes
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Do A and A^T have the same eigenvectors?
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No
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Do A and A^T have the same eigenvalues?
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No, but they have the same eivenvectors
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Do similar matrices ahve the same eigenvalues?
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Yes
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What is the kernel if rank = m?
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0
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What is the kernel if image is not equal to n?
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0
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What is the kernel if det = 0?
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0
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Are all invertible matrices necessairly similar?
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No
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What is the relationship between eivenvectors of symmetric matrices?
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The eigenvectors are perpendicular
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For a 2•2 matrix, what conditions have to be met in order to be stable?
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tr is less than 0
det is greater than 0 |
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For a discrete case, when is the matrix stable?
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If eigenvalues = p +/- iq and sq. rt.(p^2 + q^2) is less than one
The absolute value of each eigenvalue must be less than one |
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What is the least squares solution contained within?
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Im(A) perpendicular
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Can rref change determinant?
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Yes, except when it's not invertible. In that case, the determinant is = to 0 regardless
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Can rref change eigenvalues and eigenvectors?
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Yes
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Can rref change kernel?
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No
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Are upper triangular matrices necessarily diagonizable?
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No
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What is the relationship between the determinants of simliar matrices?
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They are the same
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Can you have the same determinant and different eigenvalues?
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Yes
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What is the relationship between the eigenvalues of similar matrices?
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They are the same.
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What is a similar matrix?
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Symmetric with the same eigenvalues
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If an nxn matrix has a nontrivial kernel, what is guaranteed to be an eigenvalue?
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0
Av=0v |
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What does it mean in terms of eigenvalues if nxn has a rank less than n?
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0 is an eigenvalue, because kernel is nontrivial
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If kernel is nontrivial, what is guaranteed to be an eigenvalue?
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0
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If A is diagonizable, does A^2 have to be diagonizable?
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No
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In order to be similar, what conditions have to be met for geometric and algebraic multiplicities?
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They have to be the same.
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If A^2 = A, what does it mean in terms of eigenvalues?
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The eigenvalue is either 0 or 1, because A^2v = lambda^2 v = lambda v = Av
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Do A^T and A have the same determinant?
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Yes, as well as the same eigenvalues, but not the same e
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Do A^T and A have the same eigenvalue?
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Yes, as well as the same determinant, but not the same e
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When is te unique squares solution uniqu?
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When ker is non-trivial, because if v is in ker, then x•nv is a solution
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What does it mean if a matrix represents a projection onto a line?
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It's symmetric
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What are the eigenvalues of a projection?
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0 and 1
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