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...
18. By induction, prove that if A1, A , ... , A , are invertible matrices of the same size, then the product A A2... A is invertible and (A1A2 An)= AnAA 相关知识点: 试题来源: 解析 若(A_1A_2⋯A_(n-1)^(-1)=A_(n-1)^(-1)⋅A_2^(-1)A_1^(-1) 则 ...
If {eq}A {/eq} is an {eq}m \times n {/eq} matrix and {eq}B {/eq} is an {eq}n \times r {/eq} matrix, then their matrix product {eq}AB {/eq} is also a matrix but of order {eq}m \times r {/eq}. Note that the product of two square matrices of order {e...
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...
Vector Product:Vectors are quantities that have both magnitude and direction. In physics, these are represented by arrow and sometimes these are represented by matrices. An operation involving vectors is the cross product. When taking the cross product of two vectors, the result is...
How to prove that a matrix is diagonal if its columns are orthogonal? 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 a) Prove that the p...