Block diagonalized matrices

This idea can easily be extended to more than two matrices to yield a matrix with non-vanishing elements in square blocks along the main diagonal and zeros elsewhere. Such a block-diagonal matrix (e.g. D = A B C) has the self-evident important properties ... 

In this case, the variables for the primal SDP problem with free variables (Eq. (3)) and the dual SDP problem with equality constraints (Eq. (4)) are X,x) G S X IR and y G IR , respectively. Therefore the size of an SDP problem depends now on the size of each block-diagonal matrix of X, m, and s. We should also mention that the problem as represented by Eq. (4) is the preferred format for the dual SDP formulation of the variational calculation, which we present in the next section, too. 

The difficulty here is how to simultaneously constrain y and / y to be positive semidefinite. To formulate it as a primal SDP problem (Eq. (1)), we should express these two conditions as a positive semidefinite constraint over a single matrix let y be a block-diagonal matrix in which two symmetric matrices yj and y2 are arranged diagonally, and let us express the interrelation between these two matrices via linear constraints defined by the matrices Ap and the constants as in Eq. (1). That is. 

Prepare a block-diagonal matrix in which y and / — y are placed diagonally. 

Let F be a block-diagonal matrix where F and Q are diagonally arranged ... 

The determinant of a block-diagonal matrix equals the product of the block determinants (Section 1.2). Therefore (2.17) gives... 

First we observe that any matrix is similar to a block diagonal matrix, where the sub-matrices along the main diagonal are Jordan blocks. It is thus sufficient to prove that any Jordan block can be transformed to a complex symmetric matrix. In passing we note that any matrix with distinct eigenvalues can be brought to diagonal form by a similarity transformation. The key study therefore relates to XI + J (0), where 1 is the n-dimensional unit matrix and... 

By rearranging elements corresponding to To andTo oefl,..., 6 respectively, leads to the following block-diagonal matrix ... 

This can be seen from the definition (Eq. 17) because mixed p, 7t-covariance terms will then vanish. 0p is a block-diagonal matrix with blocks 0. Note that 0y does not consist of 0p alone this would only be true if the vector n were (erroneously) taken to be a constant (nonrandom) approximation (comparable to Y° see Eq. 18). 

Notice that the product of two equally block-diagonalized matrices— such as those two above—is another similarly block-diagonalized matrix. It is especially important that this resulting matrix can be obtained simply by multiplying the corresponding individual blocks of the original matrices. Check this on the above example ... 

It is possible to partially solve the conjecture in [5] in the vanishing coupling limit, where the dynamic matrix D degenerates to a block-diagonal matrix ... 

For Cantor chains, the dynamical matrix is a block-diagonal matrix, where the block matrices are taken with multiplicity 2 from the set of matrices of type ... 

Let us examine the structure of the block-diagonal matrix describing orbital interaction within the Hiickel scheme ... 

Let us first put our problem under a convenient form. If we introduce the (N x I) x (N x I)-dimensional block-diagonal matrix... 

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