scholarly journals On similarity of an arbitrary matrix to a block diagonal matrix

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1205-1214
Author(s):  
Michael Gil’
Keyword(s):  
Diagonal Matrix ◽  
Spectral Variation ◽  
Matrix Functions ◽  
Invertible Matrix ◽  
Algebraic Multiplicity ◽  
Block Diagonal Matrix ◽  
Arbitrary Matrix ◽  
Norm Estimates ◽  
Unique Eigenvalue ◽  
Block Diagonal

Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||T||||T-1|| is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

scholarly journals The Tropical Matrix Groups with Symmetric Idempotents

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Lin Yang
Keyword(s):  
Diagonal Matrix ◽  
Main Diagonal ◽  
Diagonal Block ◽  
Maximal Subgroups ◽  
Block Diagonal Matrix ◽  
Matrix Groups ◽  
Idempotent Matrix ◽  
Block Idempotent ◽  
Real Matrices ◽  
Block Diagonal

In this paper we study the semigroup Mn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.

scholarly journals Multi-Stage Convolutional Broad Learning with Block Diagonal Constraint for Hyperspectral Image Classification

2021 ◽  
Vol 13 (17) ◽  
pp. 3412
Author(s):  
Yi Kong ◽  
Xuesong Wang ◽  
Yuhu Cheng ◽  
C. L. Philip Chen
Keyword(s):  
Hyperspectral Image ◽  
Diagonal Matrix ◽  
Hyperspectral Image Classification ◽  
Block Diagonal Matrix ◽  
Multi Stage ◽  
Alternating Direction ◽  
Mapping Process ◽  
Mutual Independence ◽  
Sparse Features ◽  
Block Diagonal

By combining the broad learning and a convolutional neural network (CNN), a block-diagonal constrained multi-stage convolutional broad learning (MSCBL-BD) method is proposed for hyperspectral image (HSI) classification. Firstly, as the linear sparse feature extracted by the conventional broad learning method cannot fully characterize the complex spatial-spectral features of HSIs, we replace the linear sparse features in the mapped feature (MF) with the features extracted by the CNN to achieve more complex nonlinear mapping. Then, in the multi-layer mapping process of the CNN, information loss occurs to a certain degree. To this end, the multi-stage convolutional features (MSCFs) extracted by the CNN are expanded to obtain the multi-stage broad features (MSBFs). MSCFs and MSBFs are further spliced to obtain multi-stage convolutional broad features (MSCBFs). Additionally, in order to enhance the mutual independence between MSCBFs, a block diagonal constraint is introduced, and MSCBFs are mapped by a block diagonal matrix, so that each feature is represented linearly only by features of the same stage. Finally, the output layer weights of MSCBL-BD and the desired block-diagonal matrix are solved by the alternating direction method of multipliers. Experimental results on three popular HSI datasets demonstrate the superiority of MSCBL-BD.

AN ALGEBRAIC PROBLEM ARISING IN BIOMATHEMATICS

2016 ◽  
Vol 78 (6-6) ◽  
Author(s):  
Sisilia Sylviani ◽  
Ema Carnia ◽  
A. K. Supriatna
Keyword(s):  
Population Growth ◽  
Matrix Model ◽  
Special Form ◽  
Matrix Multiplication ◽  
Diagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonal Matrix ◽  
Algebraic Problem ◽  
The Matrix ◽  
Block Diagonal

This paper discusses a matrix model that describes the dynamics of a population with m live stages and lives in n patch seen from algebra viewpoint. The matrix D describes population growth in a patch or location. The matrix D is defined as a matrix obtained from matrix multiplication of a permutation matrix with a block diagonal matrix that its diagonal blocks is matrices with non-negative entries and transpose of a permutation matrix [4]. It will be shown that the permutation matrix contained in D has a special form.

scholarly journals Completing a block diagonal matrix with a partially prescribed inverse

1995 ◽  
Vol 223-224 ◽  
pp. 73-87 ◽  
Author(s):  
Wayne W. Barrett ◽  
Michael E. Lundquist ◽  
Charles R. Johnson ◽  
Hugo J. Woerdeman
Keyword(s):  
Diagonal Matrix ◽  
Block Diagonal Matrix ◽  
Block Diagonal
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