Prove that a square complex matrix is unitary if and only if its columns vectors form an orthonormal set in C", (n e Z+).

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

A:

Q: Let S be an n × n singular matrix. Prove that for any n × n matrix B, the matrix SB is also…

A: Singular matrix :A matrix is singular if and only if it has a determinant of 0

Q: ) Find all scalars A such that A(2,3) has length 1.

A: Given query is to find the value of scalars for which length of given vector is 1.

Q: 2. Prove that: k(4+ mB) - kA+(km) B for any mx n matrices A and B, and scalars k and m. (You may not…

A: 2 We have to prove that kA+mB=kA+kmB Where, A and B are m×n matrices and k and m are the scalars.…

Q: Compute the product 0 () 0 (²) as a 2 x 2 matrix. What kind of isometry is this product?

A: Given product is  Oπ5O2π15 where Oθ represents rotation matrix through an angle θ in anti-clockwise…

Q: Prove that a 2 ×2 matrix A is reducible if and only if a12a21 = 0.

A:

Q: Define a complex matrix B to be skew-Hermitian if Bh = −B. Show that if B is a complex matrix that…

A: We are given that  B is a skew-harmattan matrix . And basically we have to show that 'every…

Q: Compute the adjugate of A = the cofactor matrix of A. -2 0 3 5 -2 -2 1 2 noting that adj(A) =…

A:

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

A:

Q: - Lat A = - 2 - 2 3. 2. 1- - 2 11

A:

Q: Prove that the product of two unitary matrices is unitary

A: Suppose we have two unitary matrices A and B So by properties of unitary AAT= I   ...........(1) BBT…

Q: (a) Verify that the matrix cos o - sin o sin o cos o is orthogonal. (b) Prove that if A is an…

A: "Since you asked multiple questions, we will solve the first question for you. If you want any…

Q: Assume that A and B are in Ma(C). Show that if Bis invertible, then there exists a scalar e e C such…

A:

Q: Suppose A is a complex matrix. Show that AA and A" A are Hermitian.

A: Explanation of the answer is as follows

Q: Prove that cond(A) = 1 if and only if A is a scalar multiple of a unitary or orthogonal matrix.

A:

Q: Let C be a k x k symmetric matrix. Then, which of the following is not true? a) C is diagonalizable.…

A: Diagonalization of the matrix and true false.

Q: Let A be a nonsingular n×n matrix with a nonzero cofactor Ann, and set c = det(A)/Ann Show that if…

A: Let A is any non singular nxn matrix with non zeroCofactor Ann and Set C = det(A)Ann   By concept…

Q: 4. Prove that det (A) = det (A¹) for a 2 × 2 matrix.

A:

Q: Prove that the n X n identity matrix In is invertible and that

A: We have to prove that the n X n identity matrix In is invertible and that In-1=In.

Q: Prove that an upper triangular n ×n matrix is invertible if and only if all its diagonal entries are…

A:

Q: Suppose that A is a 10-by-10 matrix over the complex numbers. Suppose that the singular values of A…

A: Please check the next step for details

Q: (a) Show that the Jordan canonical form of an invertible matrix A is an upper triangular matrix J…

A:

Q: | Verify that Hamilton’s relations hold for the matrices 1, i, j, and k. Also show (assuming…

A: Note:- In given problem, exact one question is not mentioned  (that is, problem is yellow marked  or…

Q: Prove that if a square complex matrix is unitarily diagonalizable, then it must be normal.

A: Every square complex unitarily diagonalizable matrix is normal. Let, A be a square complex unitarily…

Q: Let B be an invertible matrix. Show that A = BTB is positive definite.

A:

Q: 5. Prove that the cigenvalues of a Hermitian matrix are always real-valued, and that the inner…

A: Both the proofs of the given question has been proved in step 2 and step 3.

Q: Compute the characteristic polynomial of the matrix c-(:) Hence show that C has no non-zero…

A:

Q: Let A be mxh matrix. an Prove that. AlL %3D max

A: (Solving the one question as per our guidelines) Solution: Given A is an m×n matrix. We have to…

Q: Given a matrix where α and β are non-zero complex numbers, find its eigenvalues and eigenvec- tors.…

A:

Q: Suppose that U is a 3-by-3 unitary matrix. Prove that U is a normal matrix and prove that there…

A: It is given that U is a 3×3 unitary matrix. Prove that U is a normal matrix. Show that there exists…

Q: 13. Find all 2 x 2 matrices A = (" ta² = (C :) such that %3D

A: We need to find the set of all 2×2 matrices such that A2=O, where O is the zero matrix of order 2.

Q: FIND THE FUNDAMENTAL MATRIX AS W ELL AS THE FUNDAMENTAL MATRIX SATISFYING )- I x'= -4 %3D

A: For the solution follow the next steps.

Q: Prove that if A is an n x n real and symmetric matrix, then eigenvectors corresponding to different…

A:

Q: 2 Find the cosine of the angle between these two matrices, using the standard matrix inner product…

A: Solution

Q: Use triangular matrix method to find det(8). DO NOT use co-factor expansion -2 -3 -2 1

A: We need to find the determinant of the matrix B=1-11001041-2123-3-21 Let R1, R2 ,R3,  R4 denote the…

Q: If v is an eigenvector of A, then ū is an eigenvector of A".

A:

Q: An n x n matrix A is invertible if ВА In АВ II

A: Inverse Matrix: Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I = BA,…

Q: (6) Consider the real symmetric matrix (2 1 0 A 1 1 1 1 2 Construct an orthogonal matrix Q of eigen…

A:

Q: A Hessenberg matrix H, is n x n matrix that has 2 on the main diaonal, 1 everywhere below the main…

A:

Q: 5. Prove that the matrices A and A' have the same eigen values.EK

A:

Q: Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is…

A: Consider A be a lower triangular matrix with non-zero entries on the diagonal.

Q: If A is an eigenvalue of A, then A is an eigenvalue of AH.

A:

Q: 7. Suppose S is a normal operator on a finite-dimensional complex inner product space, all of whose…

A: If V be a complex inner product space and S is a a normal operator on V then SS*=S*S A operator S is…

Q: Suppose that  β and γ are ordered bases for an n-dimensional real [complex] inner product space V.…

A: Let us write the ordered basis as.

Q: (b) Show that an n x n complex matrix is unitarily diagonalizable if and only if it is normal.

A: By using Schurs theorem the answer is given as follows :

Q: Suppose that {v1, v2,..., vn} is an orthonormal set in Rn with the standard inner product. Let the…

A: Suppose that {v1 v2 v3 ---vn} be an orthonormal set in Rn with the standard inner product. Let the…

Q: - Suppose that A and B arenxn matrices and A is invertible. Invent and prove a theorem that…

A: Solution

Q: Suppose the eigenvalues associated with a 2 by 2 matrix of a system of linear ODE's are complex…

A:

Q: 22. Prove that a square matrix A is not invertible if and only if zero is an eigenvalue of A.

A:

Q: Which of the following is true or false? a)An orthogonal and symmetric matrix Q satisfies Q2=I…

A: a Since Qis orthogonal matrix. ⇒QTQ=I ...i and Q is symmetric ⇒QT=Q ...ii Now, substitute the value…