Let A and B be Hermitian matrices of the same order. Show that AB is Hermitian if and only if AB BA =.Proof If A B B A =, then ()()H H H H AB BA A B AB ===. Hence, AB is Hermitian.Conversely, if AB is Hermitian, then ()H AB AB =. Therefore, H H AB B A ...
2.4.6 Nonderogatory Matrix A matrix A is nonderogatory if and only if it is similar to a companion matrix of its characteristic polynomial. That is, A is a nonderogatory matrix if and only if there exists a nonsingular matrix T such that T−1 AT is a companion matrix. ...
If A and B are symmetric matrices, prove that AB - BA is a skew-symmetric matrix. Prove that any skew-symmetric matrix is square. Suppose A is a square matrix. Prove that A - A^T is a skew-symmetric matrix. Prove the given theorem: Any square matrix c...
14、that x" Ax is real-valued for all x e C" if and only if A is Hermitian.第9题 Let A, B G C/,xn be Hermitian matrices, and A be positive definite. Show that AB is similar to BA , and is similar to a real diagonal matrix.选做题得分若正面不够书写,请写在反面.第6题解答Le...
matrix A. We say that an n × n matrix B is an inverse for A if and only if AB = BA = I, where I is the n ×n identity matrix. The reason that we want to consider inverses for matrices is that they enable us to easily obtain solutions to linear systems of equations. If we...
Here we define and investigate classes of more restricted reversible rings which fulfill stronger commutative requirements, for example, rings that satisfy ab = 0 = ac + db implies ba = 0 = ca + bd.doi:10.1080/00927872.2013.867969Veldsman...
Show that the matrix I - A - B + AB is invertible and (I - A - B + AB)^{-1} = I + A + B + BA. 2. Prove that if A is an n \ how to show subspace of a vector space Let W1 and W2 be two subspaces of a vector space...
A square matrix A is invertible if and only if A is a non-singular matrix. Proof: Let A be an invertible matrix of order n and I be the identity matrix of the same order. Then there exists a square matrix B of order n such that AB = BA = I. Now, AB = I. So |A| |B| ...
What if a 2 \times 2 matrix has only one eigenvalue? Let A := ( 1 3 0 2 ) B := ( 2 2 1 5 ) Find a 2 2 matrix X for which the following matrix equation is satisfied: A X B = A T B 1 + A B Suppose B is a 2 x 2 matrix satisfying AB = B + I, wher...
The trace of ann×nmatrix is the sum of itsdiagonal elementsaii, 1≤i≤n, ortraceA=∑i=1naii. The trace occurs in many matrix formulas, and we will encounter it in later chapters. It is important to note that even thoughAB≠BAin general, in fact trace (AB)=trace (BA). ...