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36 Cards in this Set

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