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.)

Provide a recursive definition of some sequence of numbers or function (e.g. log, exponent, polynomial). Choose...

Provide a recursive definition of some sequence of numbers or function (e.g. log, exponent, polynomial). Choose one different from that of any posted thus far. Write a recursive method that given n, computes the nth term of that sequence. Also provide an equivalent iterative implementation. How do the two implementations compare?

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