An orthogonal matrix {eq}X {/eq} has the property that {eq}X^tX=I {/eq}. Prove that {eq}X^tAX \text{ and } A {/eq} have the same characteristic polynomial. Characteristic Polynomials of Matrices First we are given an orthogonal...
Note that the product of two square matrices of order n×n remain an n×n matrix. Answer and Explanation: First, calculate the two products, starting with AB: $$\begin{align*} AB &= \begin{bmatrix} a_{11} & a_{12} \ a_{21} &......
Problem 3. Given a fixed ()-matrix , define a transformation : via () = (where, as usual, and are the spaces of ()- and, respectively, ()-matrices). Prove that rk = rk . (: Ker admits an easy explicit description in terms of the null space of .)Solution: To find Ker one ...
Prove that diagonal matrices are symmetric matrices. Definition of Diagonal Matrices and Symmetric Matrices: Diagonal Matrices: A square matrix is called a diagonal matrix if it has 0 on all positions except the main diagonal. For example: Identity matrices of all orders are diagonal matrices. ...
2. Prove that if A is an n \ How to prove that eigenvalues are those of a matrix? Prove that given two m x n matrices A=[a_ij] and B= [b_ij] prove that A+B=B+A. Compute the inverse to the...
Symmetric Matrices: A matrixMis a symmetric matrix if the transpose of the matrix is equal to the matrix itself. This implies thatM=MT. It is known that the matrixMThas ordern×mif the matrixMis of orderm×n. So forM=MT, the dimensions must be equal, that is,m×n=n×m. ...
Prove that given two m x n matrices A=[a_ij] and B= [b_ij] prove that A+B=B+A. Compute the inverse to the matrix A below How to determine if matrix is invertible? Does every invertible matrix have n eigenvalues? Prove that two eigenvectors of a real symmetric matrix, if they co...
Prove the following: If A and B are unitary operators, then the product C = AB is also a unitary operator. If A and B are Hermitian operators, prove that the operator AB is Hermitian if A and B commute (AB = BA). Prove the following results involv...
Prove that if U and W are vector subspaces of a vector V, then UW is also a vector subspace of V.Products of Vector Spaces:When we have two subspaces and we consider their product, it can easily be shown that the resultant space is a vector space. Howev...
Prove that an upper or lower triangular n×n matrix is invertible if and only if all its diagonal entries are nonzero. Triangular Matrices: A matrix that has the value 0 in all the places above the main diagonal is the lower triangular ...