# Relationship between determinant and norm

### Linear Algebra: Eigenvalues, Determinant, and Trace

Not in the sense of norms. The determinant of a matrix does not satisfy the axioms of a normNorm (mathematics) - Wikipedia, which is often. The svds function returns a vector of sorted singular values, so the top value is the largest singular value of matrix M. 5. Use the norme function to find the. Limit Theorems; Modes of Convergence · Relationship of Modes · DCT . The definition above states that the determinant is a sum of many terms, each a . The determinant of a triangular matrix is the product of its diagonal elements. .. The Frobenius norm of a matrix A is the Euclidean norm of the vector.

The skew circulant matrices as preconditioners for linear multistep formulae- LMF- based ordinary differential equations ODEs codes. Hermitian and skew-Hermitian Toeplitz systems are considered in [ 12 — 15 ].

Spectral decompositions of skew circulant and skew left circulant matrices were discussed in [ 17 ]. Compared with cyclic convolution algorithm, the skew cyclic convolution algorithm [ 8 ] is able to perform filtering procedure in approximate half of computational cost for real signals. In [ 2 ] two new normal-form realizations are presented which utilize circulant and skew circulant matrices as their state transition matrices.

The well-known second-order coupled form is a special case of the skew circulant form. In [ 3 ], a new fast algorithm for optimal design of block digital filters BDFs was proposed based on skew circulant matrix. Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [ 1011 ]. Unfortunately, the computational complexity of these algorithms is very amazing with the order of matrix increasing.

However, some authors gave the explicit determinants and inverse of circulant and skew circulant involving some famous numbers.

## Abstract and Applied Analysis

For example, Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [ 19 ]. Lind presented the determinants of circulant and skew circulant involving the Fibonacci numbers [ 20 ]. Dazheng [ 21 ] gave the determinant of the Fibonacci-Lucas quasicyclic matrices.

Recently, there are several papers on the norms of some special matrices. Solak [ 26 ] established the lower and upper bounds for the spectral norms of circulant matrices with the classical Fibonacci and Lucas numbers entries.

Shen and Cen [ 28 ] gave upper and lower bounds for the spectral norms of -circulant matrices in the forms of, and they also obtained some bounds for the spectral norms of Kronecker and Hadamard products of matrix and matrix. Akbulak and Bozkurt [ 29 ] found upper and lower bounds for the spectral norms of Toeplitz matrices such that and. By using a Strang-type block circulant preconditioner, Zhang et al.

In [ 11 ], nonsymmetric, large, and sparse linear systems were solved by using the generalized minimal residual GMRES method; a circulant block preconditioner was proposed to speed up the convergence rate of the GMRES method.

Wilde [ 12 ] developed a theory for the solution of ordinary and partial differential equations whose structure involves the algebra of circulants. He showed how the algebra of circulants relates to the study of the harmonic oscillator, the Cauchy-Riemann equations, Laplace's equation, the Lorentz transformation, and the wave equation.

And he used circulants to suggest natural generalizations of these equations to higher dimensions. Circulant matrices have important applications in various disciplines including image processing [ 13 — 15 ], communications, signal processing [ 16 ], encoding, solving Toeplitz matrix problems, preconditioner, and solving least squares problems.

They have been put on firm basis with the work of Davis [ 17 ] and Jiang and Zhou [ 18 ].

Some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [ 1718 ]. Unfortunately, the computational complexity of these algorithms is exorbitant with the order of matrix increasing. However, some authors gave the explicit determinants and inverses of circulant and skew circulant involving some famous numbers.

For example, Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [ 19 ]. Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers [ 20 ].

### linear algebra - Relations between matrix norm and determinant - Mathematics Stack Exchange

Lin [ 21 ] gave the determinant of the Fibonacci-Lucas quasi-cyclic matrices. Shen considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [ 22 ]. Recently, there are several papers on the norms of some special matrices.

Solak [ 26 ] established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries. Ipek [ 27 ] investigated an improved estimation for spectral norms of these matrices. Shen and Cen [ 28 ] gave upper and lower bounds for the spectral norms of -circulant matrices in the forms of andand they also obtained some bounds for the spectral norms of Kronecker and Hadamard products of matrix and matrix.