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...
Prove the following: If A and B are unitary operators, then the product C = AB is also a unitary operator. Prove that the characteristic roots of a Hermitian matrix are real. A and B are real non-zero 3 \times 3 matrices and satisfy the equation (AB) T + B...
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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 ...
For two matrices to get multiplied, the inner dimensions of the matrices must be same, i.e the number of columns of the first matrix is equal to the number of rows of the second matrix. Am×n C For multiplication, multiply ...
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 orthogon...
A perfect square is defined as a number which can be expressed as the product of two same integer values. For example: 16 is a perfect square number since it can be written as 4×4. An even integer can be expressed in the form of 2n, where n is an integer ...
Symmetric Matrices:A matrix M is a symmetric matrix if the transpose of the matrix is equal to the matrix itself. This implies that M=MT. It is known that the matrix MT has order n×m if the matrix M is of order m×n. So for M=MT, the dimensions must be equal, that is...