Unformatted text preview: Review about the Matrix We will have a quick look over the main definition and theorems link to the matrix. A matrix over field is a set of numbers from arranged in a rectangular grid as follows: 123 1 4 5 6 is 2 789 3 4 5 6 7 8 . 9 If we, in addition, replace the complex entries of by their complex conjugates, then A C we obtain the so called adjoint matrix , that is, A matrix satisfying is called symmetric. A matrix satisfying is called selfadjoint or Hermitian. For real matrices the notions 'Hermitian' and 'symmetric' have the same meaning. Can we consider the following matrix as a Hermitian one ? 1 2 8 2 2 4 8 4 3 The matrix A can be abbreviated: A = where i and j are respectively the row and the column index. Transposed, Adjoint, Symmetric and Hermitian matrices : Interchanging rows and columns of a matrix we obtain the so called transposed matrix , that is, For instance, the transposed matrix of It is not a Hermitian matrix due to the last term of the diagonal which must be a real number, because 3i becomes ‐3i. 1 2 The following rules hold true: determinant is 12 = 1*4 ‐3*2 = ‐2. 34 Orthogonal and unitary matrices A complex (n x n)‐matrix is called unitary if that is, if the columns of form a orthonormal . For real matrices the complex basis of conjugation can be omitted (as with the scalar product). A unitary matrix with real entries is called orthogonal matrix. The determinant : The following notation are used for the determinant of a matrix : Eigenvalues A scalar is called eigenvalue of square matrix if The vectors ν 0with are called eigenvectors associated with eigenvalue . The set of the zero vector and of all eigenvectors associated with a given eigenvalue forms a vector space called eigenspace λ of . Properties: The eigenvalues remain unchanged under similarity transformation like That is, an eigenvector of associated with eigenvalue corresponds to eigenvector of associated with the same eigenvalue. 3
4 In case of a matrix , the Characteristic polynomial The eigenvalues of matrix are the zeros of the characteristic polynomial SINGULAR VALUE DECOMPOSITION For any real (m x n)‐matrix there exist orthogonal matrices and with S consist of the square roots of the eigenvalues of in descending order. The columns of the unitary matrices U and V are eigenvectors of and , resp. We have , hence . Example of SVD : 1 0 1 1 1 0 21 12 The singular values are the square roots of the eigenvalues of , is the rank of , and the columns of and are eigenvectors of and , resp. [U,S,V] = SVD(A) calculates(matlab) the singular value decomposition of (m x n)‐matrix A. The diagonal entries of the returned (m x n)‐matrix 5 We calculate first the eigenvalues of ATA and the eigenvectors corresponding to the eigenvalues. We find λ1=3, λ2=1 and respectively we have v1=[√2/2,√2/2], v2=[√2/2,‐√2/2]. Then we have singular values which are the square root of eigenvalues and we find normed vectors u1, u2 and u3. For the two first, we have: U1=A. v1 and u2=A. v2 then for u3 we use the Gram‐Schmidt process because it needs to be orthogonal with u1 and u2. Finally we get the matrix U. 6 Concerning the dimension, it does not match up but it only does because of the original matrix which is not square. √3 0 0 1 0 0 √3 3 1 1 √2/2 01 1 0 √2/2 1 1 √2/2 01 √2/2 10 √3/3 √3/3 √3/3 :A √3/3 √6/30 √3 0 √2/2√2/2 √2/2 √3/3 0 √6/6 1 √2/2 √2/2 0 √6/6√2/2 √3/3 0 0 √2/2 √2/2 √6/3 √6/6 √6/6 Additionally In mathematics, the conjugate transpose, Hermitian transpose, or adjoint 7 matrix of an m‐by‐ n matrix A with complex entries is the n‐by‐ m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by where the subscripts denote the i, j‐th entry, for 1 ≤ i ≤ n an 1 ≤ j ≤ m, and the overbar nd denotes a scalar complex conjugate. (The complex conjugate of a + bi, where a and b are reals, is a − bi.) This definition can also be written as where denotes the transpose and denotes the matrix with complex conjugated entries. Other names for the conjugate transpose of a matrix are Hermitian conjugate, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols: or , commonly used in linear algebra 8 (sometimes pronounced "A dagger"), universally used in quantum mechanics , although this symbol is more commonly used for the Moore‐Penrose pseudoinverse In some contexts, denotes the matrix with complex conjugated entries, and thus the conjugate transpose is denoted by or . Example Trace (linear algebra) In linear algebra, the trace of an n‐by‐n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii represents the entry on the ith row and ith column of A. Equivalently, the trace of a matrix is the sum of its eigenvalues, making it an invariant with respect to a change of basis. This characterization can be used to define the trace for a linear operator in general. Note that the trace is only defined for h a square matrix (i.e. n×n). If Then 9 10 Basic remarks If the entries of A are real, then A* coincides with the transpose AT of A. A square matrix A with entries aij is called Hermitian or self‐adjoint if A = A*, i.e., ; skew Hermitian or antihermitian if A = −A*, i.e., ; *A = AA*. normal if A Even if A is not square, the two matrices A*A and AA* are both Hermitian and in fact positive semi‐definite matrices. The adjoint matrix A* should not be confused with the adjugate adj(A) (which is also sometimes called "adjoint"). 11 ...
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- Spring '10
- Hermitian, square matrix, conjugate transpose