A ∈ Rnn which of the following is true?   a) If A is regular, the inverse of A can be found with the Cayley-Hamilton theorem. b) < , > is an inner product in Rn1 and u, v ∈ Rn1 and = . c) Rn1 is the direct sum of the eigenspaces obtained versus the eigenvalues ​​of A. d) The characteristic polynomial of A divides the minimal polynomial. e) If L linear transformation has inverse and its matrix is ​​A, then L-1 linear transformation has matrix A as well.

Q: Г-4 0 21 Let A = 2 4 -8 2 0 -4 a) Find the characteristic polynomial of A. b) Find the eigenvalues…

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Q: (9) The Jordan canonical form is obtained with a basis of eigenvectors.

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Q: Consider the quadratic form F(X₁, X₂, X 3) = 25X² +34X² +41 X ² Write F - 24 X₂ X3 al F= [X² [A] [X…

A: The given quadratic form is given by  F(x1,x2,x3 ) = 25 + 34 + 41 - 24

Q: Let {u1,..., u„} be an orthonormal basis for R", and let 21, ...,A, be any real scalars. Define A =…

A: According to the given information suppose that,

Q: Compute (a) the characteristic polynomial of A, (b) theigenvalues of A, ( c) a basis for each…

A: It is given that,  A = 2110014500310002. We have to compute the following : a)  The characteristic…

Q: Consider the arrays ¡C= 30-10 0200 -1030 0002 y D= 3 -1 1 -1 -1 3 -1 1 1 -1 3 -1 1 −1 3 −1…

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Q: Define the following:          Characteristic polynomial Eigen-values and Eigen- vectors

A: Characteristic polynomial: Characteristic polynomial is associated with square matrix , it is a…

Q: Consider matrix (3 1 1 A = 4 2 E C3,3 3 3 a) Find the characteristic polynomial of A. b) values.…

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Q: Suppose that T is a self-adjoint operator on a finite-dimensional inner = 2 are the only eigenvalues…

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Q: a) Suppose that A e M, has only one distinct eigenvalue, say a, of algebraic multiplicity n. Prove…

A: We will use the fact that α is real and the diagonal matrix must be an identity matrix

Q: A- 0001 0354 3021 1000 a) Determine the characteristic polynomial of matrix A. b) Determine…

A: Note:- As per our guidelines and due to time limit, we can solve first three sub parts of this…

Q: or (Mt 0)

A: answer is in next step

Q: H is a Hermitian operator in a finite dimensional inner product space V. Show that H has at least…

A: Eigenvectors: An eigenvector v of an operator T is a non-zero vector corresponding to the eigenvalue…

Q: If A is diagonalizable and for all eigenvalues, A of A, |A| = 1, then A is unitary.

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Q: Let 2 2 = x₂ = Compute the characteristic polynomial of A. Use the variable s. PA(S) = = The…

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Q: Let x e gl(n, F) nave n distinct eigenvalues a,,..., an in F. Prove that the eigenvalues of ad x are…

A: Let B be a basis of in which x is diagonalizable. Let A be the matrix with respect to this basis.…

Q: 24. Consider the quadratic form Q(x) = -x² - 2y² - z² + 2xy - 2yz a) Find a symmetric matrix A such…

A: As you asked more than 3 sub parts , so I solve first three sub part , please  repost the remaining…

Q: Let A, B be two square matrices of the same size over a field F such that AB = BA. Ifv is an…

A: Let's start by understanding the definitions:v is the column vector obtained by stacking the columns…

Q: Prove Proposition , An n×n matrix A is invertible if and only if 0 is not an eigenvalue of A.

A: We have to prove that an n×n matrix A is invertible if and only if zero is not an eigenvalue of the…

Q: 1. Ak 1-k-k 0 kk +1 10 1 Where k = 2323232. Then: a) Ak is invertible. b) Det Ak=k. Σιλικ M c)…

A: We have to justify a,b,c,d .

Q: i) Show that if det A = 0, then A has an eigenvalue of 0. ii) Show that if A has an eigenvalue of 0,…

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Q: Let A be an n×n rea lmatrix with complex eigenvalue λ = a +ib, where b ̸= 0, and let v = r+ is be a…

A: Eigenvalue: A number λ is said to be an eigenvalue of the square matrix A if there exists a non-zero…

Q: f rank(A – cI) is k then eigenspace of corresponding to eigenvalue c spans the space Rk. True False

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Q: Let A = 3 -1 6 2 2 0 -1 6 7 7 with its entries considered as elements of the field C. Find a basis…

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Q: Let A = (a) be a n x n matrix where ai; is the entry of the matrix on row i and column j. It is…

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Q: Assume that A is a 3 × 3 matrix with characteristic polynomial (2 – t)(–1 – t)². Complete the…

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Q: H is a Hermitian operator in a finite dimensional inner product space V. Show that there exists at…

A: Trick here is to show that H has an eigenvalue and if H has an eigenvalue then by the definition of…

Q: a) A = 2 Calculate the inverse of the 25 3 matrix using the Cramer Methool b) write the…

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Q: Let A be a diagonalizable 3x3 mat rix with PAP=-I , where I is the identity mat rix. Which of the…

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Q: dx which statement about the system at = Ax is true? The characteristic polynomial of A is p(x) = λ²…

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Q: Let Pr) = r - 4r + ar+ 30 be the characteristic polynomial of 3 x 3 matrix A, where a is a real…

A: We will find out the required eigen value of matrix A.

Q: Prove that a square complex matrix is unitary if and only if its columns vectors form an orthonormal…

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Q: Suppose TEL(C³) is defined by T(21, 22, 23) = (621 +372 +423,622 +223, 723) (a) Write down the…

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Q: Let A be an n×n rea lmatrix with complex eigenvalue λ = a +ib, where b ̸= 0, and let v = r+ is be a…

A: Given that A be a real n×n matrix with complex eigenvalue λ=a+ib where b≠0 and v=r+is be a…

Q: Find the linear transformation T: P) + P1) such that T has eigenvalues -2 and 3 associated with the…

A: Please check further steps for solution.!