Articles of matrices

Derivative of the Frobenius norm with respect to a vector

I am trying to calculate the derivative of an energy function with respect to a vector $\mathbf{x}$. The energy is given by: $\psi(\mathbf{F}) = \lVert\mathbf{F}-\mathbf{I}\rVert_F^2.$ Where $\mathbf{F}$ is a square matrix that is a function of $\mathbf{x}$ (a column vector): $\mathbf{F(x)} = (\mathbf{x}\cdot\mathbf{u^T})\mathbf{A}$ $\mathbf{u^T}$ is a constant row vector and $\mathbf{A}$ is a constant square […]

Number of possible graphs from a reachability matrix?

I need to know how to work out how many possible different digraphs can be drawn from a given reachability matrix. It needs to be with the minimum number of arcs between the nodes within the graph (which i believe is 10 for the one given). If someone could explain it step by step for […]

Prove that $A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$ is not invertible

$$A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$$ I don’t know how to start. Will be grateful for a clue. Edit: Matrix ranks and Det have not yet been presented in the material.

Matrix Multiplication – Product of and Matrix

From P59 of Intro to Lin Alg, 4th Ed by Strang & P94-95 of Linear Algebra and its Apps by Lay For relief, I denote all row vectors with superscripts and column with subscripts. Define $\mathbf{A} = \left[\begin{matrix} \vec{a^1} \\ \vdots \\ \vec{a^i} \\ \vdots \\ \vec{a^m} \end{matrix}\right]_{m \times n} \& \quad \mathbf{B} = \left[\vec{b_1} […]

changing bases/rotating axes to find reflection across y=2x

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x. Hint: A sketch of v and the line may suggest an approach. I found the matrix -3/5 6/5 4/5 2/5 which seems like it gives the reflection across y=2x But my question is: is there way to do […]

Is $(B – A)(A + B)$ symmetric if $A = A^T$ and $B = B^T$?

I have a problem where I have to say if a matrix is symmetric or not, if $A = A^T$ and $B = B^T$. According to what I know, a matrix $A$ is symmetric if $A = A^T$. The specific matrix I am a bit confused about is: $$(B – A)(A + B)$$ One of […]

loewner ordering of symetric positive definite matrices and their inverse

$M_1$ and $M_2$ are symetric positive definite matrix and $M_2>M_1$ in Loewner ordering ($M_2-M_1$ is positive definite). does this imply that $M_1^{-1}>M_2^{-1}$?

How to divide polynomial matrices

If I am given two $2\times2$ polynomial matrices and I need to divide them, what are the steps I need to follow? I know I need to do right division and left division, and that the answer will have the right quotient and remainder and the left quotient and remainder. Please show me how to […]

Invariant subspaces using matrix of linear operator

I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 & 0\\ 0 & 0 & 3\\ \end{pmatrix}\ \in M_{3} (\mathbb{R}).$$ Solution: I found the 1-dimensional subspaces: They are […]

What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation to minors of $A$? I have tried some cases starting from the definition of determinant (just like the proof of the Laplace expansion), […]