Flashcards
Linear Algebra and Matrices
Cards in this deck (8)
- 1What three operations can any linear transformation be decomposed into?Every linear transformation
can be represented as a composition of three operations: a rotation operation, a
scaling operation, and another rotation operation. - 2Describe the singular value decomposition of HH = UΛ V∗* where U ∈ Nr×Nr and V ∈Nt×Nt are unitary rotation matrices and Λ ∈nr×nt is a rectangular matrix whose diagonal elements are non-negative real numbers and whose off-diagonal elements are zero.
- 3Singular ValueThe singular values of a rectangular matrix H are the square roots of the eigenvalues of HH*.
- 4Where do we see the singular values in the singular value decompostion H=UΛ V∗The diagonal elements of Λ ie. λi for i≤max(Nt, Nr) are the ordered singular values of the matrix H,
- 5Explain how we know that singular values of H are the square roots of the eigenvalues of HH*.HH* = (U)Λ (Λ)^t U*,
U is unitary
the elements on the diagonal of Λ are λi
ie. the squared singular values λi^2 are the eigenvalues of the matrix HH* and
also of H*H. - 6Unitary MatrixA matrix is called 'unitary' if its conjugate transpose U∗ is also its inverse, ie:
U*U=UU*=I (the identity) - 7RankThe rank of a matrix A is the dimension of the vector space generated by its columns, or equivalently, the maximal number of linearly independent columns of A
- 8The rank of H is equal to...precisely the number of non-zero singular values.
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- PhD Comms (Maths) Flashcards on Linear Algebra and Matrices, created by L H on 13/02/2018.
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- L H
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