Flashcards
Canonical Forms
Cards in this deck (21)
- 1Shur decomposition of AA=UTU*,
T is upper triangular
U is unitary - 2A is a n x n matrix. When is A unitary similar to a upper triangular matrix?Always. (Shur's theorem)
- 3A=UTU* What are the eigenvalues of A?The elements of the diagonal of T
- 4EQUIV: A is normal1. AA*=A*A
2. A=UΛU*, Λ is diagonal, U is unitary
3. Iff A has n orthogonal eigenvectors - 5Name 2 types of matricies which are normalHermitian (A*=A)
Unitary (A*A=I) - 6EQUIV: A is diagonizable1. A=inv(P)ΛP, Λ is diagonal
2. A has n linearly independent eigenvectors
3. A has distinct eigenvalues (n different eigenvalues) - 7DEF. Jordan canonical form
- 8Let A be a n by n matrix. When is A unitarily similar to a diagonal matrix?When A is normal
- 9Let A be a n by n matrix When is A similar to a diagonal matrix?1. if A is normal
2. n linear independent eigenvectors
3. n distingt eigenvalues - 10Properties of jordan matrices1. The number of Jordan blocks is the number of linearly independent eigenvectors of A (equiv J).
2. Algebraic multiplicity of λ is the sum of the dimensions the jordan blocks with λ.
3. Geometric multiplicity of λ is number of jordan blocks (number of independent eigenvectors - 11Defective eigenvalue1. Appears in a jordan block of dim greater than 1
2. Algebraic multiplicity exceeds geometric multiplicity - 12defective matrixhas a defective eigenvalue
- 13Index1. Order of the largest Jordan block corresponding to λ
2. Smallest k for which rank((A-λI)^k) is minimum. - 14Number of jordan blocks of size krank (A-λI)^{k-1} + rank (A-λI)^{k+1} -2*rank (A-λI)^k
- 15Let inv(X)AX=J be the Jordan canonical form of A. What are the eigenvectors of A?columns of X at position 1, 1+m_1, 1+m_1+m_2, ..., 1+m_1+...+m_{p-1}
- 16Generalized eigenvectors of AColumns of the transformation X in the jordan canonical form of A, which does not correspond with eigenvectors
- 17Jordan chainThe set of a eigenvector and the subsequent generalized eigenvectors up until the next eigenvector.
- 18Let inv(X)AX=J be the Jordan canonical form of A. Find generalized vectors(A-λ_1I)^j x_j = 0,
where x_j is the j'th generalized vector - 19Let p be the characteristic polynomial of A. What is p(A)?p(A)=0
- 20Define minimal polynomialThe polynomial q of lowest degree such that q(A) = 0
- 21A is n by n with distinct eigenvalues. What is the minimal polynomial?
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- Matrix Analysis (Theory of eigensystems) Flashcards on Canonical Forms, created by Simon Majgaard on 08/11/2020.
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- Simon Majgaard
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