If an n× n n × n symmetric A is positive definite, then all of its eigenvalues are positive, so 0 0 is not an eigenvalue of A A. Therefore, the system of equations Ax= 0 A x = 0 has no non-trivial …

Dec 11, 2016 · An invertible matrix is one that has an inverse. The inverse itself is a matrix. Note that invertible is an adjective, while inverse (in this sense) is a noun, so they clearly cannot be …

@John: If you've learned how to solve a system of linear equations (represented by a matrix), or equivalently, how to find the inverse of a matrix, you know Gauss-Jordan elimination. If this …

Hint: Show that a certain series converges in the norm $\|\cdot \|$ and that this is an inverse for $I-A$.

Jan 26, 2014 · I am confused about the relationship between the invertibility of a matrix and its eigenvalues. What do the eigenvalues of a matrix tell you about whether a matrix is invertible …

Mar 29, 2017 · It is worth noting that there also exist diagonalizable matrices which aren't invertible, for example $\begin {bmatrix}1&0\\0&0\end {bmatrix}$, so we have invertible does …

This question is crucially missing any hypothesis about the field over which matrices are considered. The case of complex matrices is very different from the case of real matrices, and …

In Exercises 37-38, determine whether A A is invertible, and if so, find the inverse. [Hint: Solve AX = I A X = I for X X by equating corresponding entries on the two sides. 37. A =⎡⎣⎢1 1 0 0 1 1 1 0 …

Jan 19, 2021 · You are correct that all non-square matrices are non-invertible. This is why the term "singular" is reserved for the square case: the colloquial meaning of "singular" is …

Jan 15, 2022 · In the book "No BS Linear Algebra," the author states that "for linear transformations to be invertible, it only needs to be either injective or subjective," mentioning …