Show that any polynomial over C (the complex numbers) is the characteristic polynomial of some matrix...

Show that any polynomial over C (the complex numbers) is the characteristic polynomial of some matrix with complex entries. Please use detail and note any theorems utilized.

Expert Solution

Colby Messinger answered 2 years ago

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial...

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial f(x) belongs to F[x] such that f(A) ＝0.

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix...

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix computation to verify that χA(A) = 0. (4 3 -1 1) (2 1 -1 0 3 0 0 -1 2) (3*3 matrix) 2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ. A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3 A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2...

show that for any n the matrix ring M_n(F) is simple over a field F. show...

show that for any n the matrix ring M_n(F) is simple over a field F. show your work. Do not use quotient rings!

The 3 x 3 matrix A has eigenvalues 5 and 4. (a) Write the characteristic polynomial...

The 3 x 3 matrix A has eigenvalues 5 and 4. (a) Write the characteristic polynomial of A. (b) Is A diagonalizable ? Explain your answer. If A is diagonalizable, find an invertible matrix P and diagonal matrix D that diagonalize A. Matrix A : 4 0 -2 2 5 4 0 0 5

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show...

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show that if B is a nonsingular square matrix, then BTB is an SPD matrix.(Hint. you simply need to show the positive definiteness, which does requires the nonsingularity of B.)

show that a 2x2 complex matrix A is nilpotent if and only if Tr(A)=0 and Tr(A^2)=0....

show that a 2x2 complex matrix A is nilpotent if and only if Tr(A)=0 and Tr(A^2)=0. give an example of a complex 2x2 matrix which is not nilpotent but whose trace is 0

According to the Fundamental Theorem of Algebra, every nonconstant polynomial f (x) ∈ C[x] with complex...

According to the Fundamental Theorem of Algebra, every nonconstant polynomial f (x) ∈ C[x] with complex coefficients has a complex root. (a) Prove every nonconstant polynomial with complex coefficients is a product of linear polynomials. (b) Use the result of the previous exercise to prove every nonconstant polynomial with real coefficients is a product of linear and quadratic polynomials with real coefficients.

Logic & Sets (Proofs question) Show that complex numbers cannot be ordered in a way that...

Logic & Sets (Proofs question) Show that complex numbers cannot be ordered in a way that satisfies our axioms. Axioms for order: 1. if x is less than/equal to y and w is greater than zero, then wx is less than/equal to wy 2. for w, x, y, z w is less than/equal to x, y is less than/equal to z then w + y = x + z if and only iff w = x and y = z

Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

All polynomial functions are continuous over all real numbers true of false rational functions are discontinuous...

All polynomial functions are continuous over all real numbers true of false rational functions are discontinuous where the numerator is equal to zero true or false exponential functions are continuous over all real numbers true or false log functions are continuous over all real numbers true or false The first derivative of a function gives the average rate of change at a point true or false The second derivative of a function gives the instantaneous rate of change at a...

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