Suppose that  β and γ are ordered bases for an n-dimensional real [complex] inner product space V. Prove that if Q is an orthogonal [unitary] n ×n matrix that changes γ-coordinates into β-coordinates, then β is orthonormal if and only if γ is orthonormal.

Q: Let A be a square matrix. Prove an alternate form of the polar decomposition for A: There exists a…

A: Proof:From Singular Value Decomposition of A, we have

Q: Prove that the row vectors of an nxn invertible matrix A form a basis for R".

A: Rows of A form a basis

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: Let 9. A = 3 9 4 6. -3 2 -2 Find an orthonormal basis of the kernel of A.

A:

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

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Q: Prove that the n × n identity matrix is orthogonal.

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: Find a basis for the nullspace of the matrix. (If there is no basis, enter NONE in any single cell.)…

A: For the solution follow the next step.

Q: the jordon Canonical Form of A

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Q: Algbra prove that A is invertible if AT(a transpose)+A is a positive definite matrix.

A: Given that If A^T+A is a positive definite matrix For a positive definite matrix Det(A^T+A)>0. So…

Q: Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1,…

A: Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1,…

Q: 5. Give an example of a matrix A such that im(A) is in R2 is spanned by the vector A = %3D

A: Given that, A is a matrix such that imA is in ℝ2 is spanned by the vector 52

Q: If A and B are Hermitian matrices,prove that AB-BA is a Skew Hermitian

A:

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

A:

Q: Let 1 -5 -4 A = 1 1 -1 -3 -5 Find an orthonormal basis of the column space of A. }

A: Solution:- It is given that          A=1-5-411-1-3-52

Q: Find a basis of the linear space V of all 3 x 3 matrices A for which both | - A 1 and are…

A: We have to find the basis of linear space V.

Q: Prove that k(x,x)=xAx' is a valid kernel, where A is a symmetric positive semidefinite matrix.

A: Let  we consider      kx,x' = xTAx'…

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

A:

Q: Suppose S is a real symmetric matrix, H is a complex Hermitian matrix, P is a real orthogonal…

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: Let O(2, R) be the set of 2 x 2 orthogonal matrices. Describe all of the O(2, R)-conjugacy classes.

A: (1) Two elements x and y of a group G are said to be conjugates if there exists an element g in the…

Q: Let 1 5 A 3 3 -6 2 0 5 Find an orthonormal basis of the column space of A.

A:

Q: Let A be an n ×n matrix with complex entries. Prove that AA∗= I if and only if the rows of A form an…

A: Consider the given data and define all terms as,

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 A be an n×n matrix whose characteristic polynomial splits. Prove that A and At have the same…

A: For any matrix,

Q: If an nxn matrix is invertible, then the column vectors form a basis for R

A:

Q: Show that it is impossible to find a basis for the vector space ofn x n (n > 1) matrices such…

A: To Show: It is not possible to find a basis for the vector space of n x n (n > 1) matrices such…

Q: Let A be a 4 x 4 matrix. Present systematically all cases for the Jordan Canonical Form of A.

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: 1. Let M,(R) be the vector space of n x n matrices with real entries and let {A € M3(R) | Aª = -A}.…

A:

Q: give an example of an infinite dimensional vector space V over complex numbers and a linear operator…

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Q: Prove that if A is a positive definite matrix and A = UΣV* is a singular value decomposition of A,…

A: A  is a positive definite matrix and

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

A:

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

A:

Q: Let A be a Hermitian operator and {l0),..., n – 1)} be the orthonormal basis of the ket space…

A:

Q: Let 1 -4 A = -2 2 6 3 10 Find an orthonormal basis of the column space of A.

A:

Q: Let 1 A = 9 Find an orthonormal basis of the row space of A

A:

Q: Let A be a diagonalizable matrix and let X be the diagonalizing matrix. Show that the column…

A: Given A is a diagonalizable matrix and let X be the diagonalizing matrix.

Q: Let A be a symmetric matrix and P be an orthogonal matrix. Show that A is positive semi-definite if…

A:

Q: Let 3 A = 1 2 -8 2 Find an orthonormal basis of the row space of A.

A:

Q: × 4 matrix such that

A: In the given question, the concept of reduced row echelon form is applied. Reduced Row Echelon Form…

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: Prove that if A and B are unitarily equivalent matrices, then A is positive definite [semidefinite]…

A:

Q: 1. Consider the set of all 3x3 matrices. Prove that with standard addition and standard…

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Q: |Let -2 1 -3 A 7 -1 Find an orthonormal basis of the row space of A.

A: Use Gram -Schmidt Orthogonalization process to find the orthonormal basis of the row space of A.

Q: Prove that if L is an invertible lower triangular matrix, then L-l is also lower triangular.

A: handwriting solution