Consequently, it makes sense to discuss them being positive or negative. Note that all the eigenvalues are real because it’s a symmetric matrix all the eigenvalues are real. Question 6: Can we say that a positive definite matrix is symmetric?Īnswer: A positive definite matrix happens to be a symmetric matrix that has all positive eigenvalues. An important point to understand is that not all symmetric matrices are invertible. As such, any matrix, whose multiplication takes place (from the right or the left) with the matrix in question, results in the production of the identity matrix. Question 5: What is meant by the inverse of a symmetric matrix?Īnswer: The inverse of a symmetric matrix happens to be the same as the inverse of any matrix. Therefore, for a matrix to be skew symmetric, A’=-A. In case the transpose of a matrix happens to be equal to the negative of itself, then one can say that the matrix is skew symmetric. Question 4: Explain a skew symmetric matrix?Īnswer: A matrix can be skew symmetric only if it happens to be square. Thus, (AB – BA) is a skew-symmetric matrixĪnswer: Symmetric matrix refers to a matrix in which the transpose is equal to itself. Where C is the square matrix that we want to decompose, C T its transpose, and finally S and A are the symmetric and antisymmetric matrices respectively into which matrix C is decomposed.īelow you have a solved exercise to see how it is done.Question 2: Say true or false: If A & B are symmetric matrices of same order then A B − B A is symmetric.Īnswer : Given, A and B are symmetric matrices, therefore we have:Ĭonsider, (AB – BA)’ = (AB)’ – (BA)’…………… The inverse matrix is B 1 ( A T A) 1 A 1 A T. The reason is that the pivots of B are always at the main diagonal: see the first reference. The inverse can of B can be determined by employing our special matrix inversion routine. The formula that allows us to do it is the following: Multiply A on the left with A T, giving B A T A. Decomposition of a square matrix into a symmetric and an antisymmetric matrixĪ property that all square matrices have is that they can be decomposed into the sum of a symmetric matrix plus an antisymmetric matrix.
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