jsgoodfella said:If we take an nxn diagonal matrix, and multiply it by an nxn matrix C such that AC=CA, will C be diagonal? I know, for instance, if C is a matrix with ones in every entry, AC=CA holds. But is there a more general way to format such a counterexample, or have I already provided a sufficient "proof"?
Thanks in advance. This isn't a homework question.
phyzguy said:When you disprove a statement by providing a counter-example, one counter-example is sufficient. There is no need to provide more.
DonAntonio said:Yes, of course. Whom are you addressing and why?
DonAntonio
phyzguy said:The OP asked whether he had already provided a sufficient proof. The answer is yes - since had already provided one counter-example, this is sufficient to disprove the original statement. That's all I'm saying.
A product of diagonal matrices is a mathematical operation where two or more diagonal matrices are multiplied together to form a new matrix. This operation is commonly used in linear algebra and has various applications in fields such as physics, engineering, and computer science.
The product of diagonal matrices is calculated by multiplying the corresponding elements of each matrix. For example, if we have two diagonal matrices A and B, the product AB would be equal to a new matrix C, where each element C[i,j] is equal to the product of A[i,i] and B[i,i]. In other words, the product of diagonal matrices is the same as the product of their diagonal elements.
The product of diagonal matrices has several properties that make it useful in mathematical operations. These include the commutative property (AB = BA), the associative property (A(BC) = (AB)C), and the distributive property (A(B + C) = AB + AC). Additionally, the product of two diagonal matrices is also a diagonal matrix.
The product of diagonal matrices has various applications in different fields. In physics, it is used to calculate moments of inertia and to represent the rotation of rigid bodies. In engineering, it is used in solving systems of linear equations and in signal processing. It is also used in computer graphics to manipulate images and in machine learning for feature selection and data compression.
The main difference between the product of diagonal matrices and the product of non-diagonal matrices is that the product of diagonal matrices results in a diagonal matrix, while the product of non-diagonal matrices does not necessarily result in a diagonal matrix. Additionally, the product of diagonal matrices is usually easier to calculate since it only involves multiplying the diagonal elements, while the product of non-diagonal matrices involves multiplying all the elements of the matrices.