Displaying similar documents to “Latent roots of lambda-matrices, Kronecker sums and matricial norms”

Truncations of Gauss' square exponent theorem

Ji-Cai Liu, Shan-Shan Zhao (2022)

Czechoslovak Mathematical Journal

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We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer n , k = 0 n ( - 1 ) k 2 n - k k ( q ; q 2 ) n - k q k + 1 2 = k = - n n ( - 1 ) k q k 2 , where n m = k = 1 m 1 - q n - k + 1 1 - q k and ( a ; q ) n = k = 0 n - 1 ( 1 - a q k ) .

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) i = 1 n X i - d i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i M ( Z ) , i=1,...,n.

Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding

Jiří Rohn (2007)

Applications of Mathematics

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For a real square matrix A and an integer d 0 , let A ( d ) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A ( d ) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists...

A method for determining constants in the linear combination of exponentials

Jiří Cerha (1996)

Mathematica Bohemica

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Shifting a numerically given function b 1 exp a 1 t + + b n exp a n t we obtain a fundamental matrix of the linear differential system y ˙ = A y with a constant matrix A . Using the fundamental matrix we calculate A , calculating the eigenvalues of A we obtain a 1 , , a n and using the least square method we determine b 1 , , b n .

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an 2 n × 2 n Brualdi-Li tournament matrix B 2 n . We show that B 2 n has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of B 2 n is also determined. Related results obtained in previous articles are proven to be corollaries.