May 16, 2018 · A matrix is a compact but general way to represent any linear transform. (Linearity means that the image of a sum is the sum of the images.) Examples of linear transforms are …

Let M be a symmetric n × n matrix. Is there any equality or inequality that relates the trace and determinant of M?

So, the connection is that both methods are different expressions of the same underlying linear algebra principles. To see it more concretely, note that the adjoint matrix is the transpose of the cofactor …

Nov 17, 2017 · In fact, a quick check on Wolfram|Alpha shows that for a 2x2 matrix to be normalizable, the top left index must exactly equal the negative of the bottom right index (among other conditions) …

Apr 19, 2021 · The fastest way to compute the eigenvalues in this case is to recognize that this matrix is a rank 1 update of a multiple of the identity matrix.

Feb 17, 2020 · A matrix/system of equations is singular is there are infinite solutions, but iff there is a unique solution then its non-singular? I haven't learned how to take a determinant yet. However, my …

Apr 28, 2021 · A Markov transition matrix has all nonnegative entries and so by the Perron-Frobenius theorem has real, positive eigenvalues. In particular the largest eigenvalue is 1 by property 11 here. …

Any vector subject to the identity matrix will give you the same vector back. That is the property of the identity matrix. But in light of eigenvectors and eigenvalues, this also matches the situation for …

Matrix multiplication is associative, so you can do it in whichever order you like. You can prove it by writing the matrix multiply in summation notation each way and seeing they match.

Nonnegative matrix A A has the largest eigenvalue λ1 <1 λ 1 <1. Then, the book says (I−A)−1 (I A) 1 has the same eigenvector, with eigenvalue 1/(1−λ1) 1 / (1 λ 1). Why? Is there any other formulas …