# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

82,421 questions
36 views

10 views

### If colums of an $n\times n$ matrice are linearly independent then columns of transpose of this matrice are linearly independent.

My question is a very short: ''If colums of an $n\times n$ matrice are linearly independent then columns of transpose of this matrice are linearly independent'' Is this true? Can you explain? Thanks....
47 views

### Property of some determinant [on hold]

Prove that $$\begin{vmatrix} 1+a^2-b^2 & 2b & -2b \\ 2ab & 1-a^2+b^2 & 2a\\ 2b & -2a^2 & 1-a^2-b^2\\ \end{vmatrix}$$ is a perfect cube.
64 views

32 views

26 views

### Some doubts regarding the evaluation of the rank of a matrix

Here is a list of very novice questions that came across while studying: Suppose $A$ is an $m \times n$ matrix. Is the rank of $A\leq \min\{m,n\}$? Attempt: The "rank" of a matrix gives us the idea ...
41 views

97 views

### Is a square zero matrix positive semidefinite?

Does the fact that a square zero matrix contains non-negative eigenvalues (zeros) make it proper to say it is positive semidefinite?
27 views

29 views

### Find an invertible matrix and a diagonal matrix

I have the following question. I know what the answer is supposed to be (I put it in mathlab). However, when I go through the individual steps I keep getting the wrong answer. Can someone tell me ...
43 views

### Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.

Let $A$ be a matrix $n \times n, n \geq 2$. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
9 views

### How to write this matrix multiplication in expression using sums

So I have to take a derivative of ANS with respect to $\lambda$ so (I think) have to write it in summation form. ANS = $y'UD(D^{2} + \lambda I_{p})^{-2}DU'y$, where y (1 x n), U (n x p), D (p x p). ...
20 views

30 views

### Showing that $I - \alpha P$ has no zero eigenvalues when $P$ is a Stochastic matrix

I am trying to to show: $$I - \alpha P$$ is non-singular and $\alpha \in [0,1)$. I know how to do it using nullspace arguments (see this) but I wanted to do it by showing the eigenvalues are NOT ...
11 views

25 views

### how to find a basis for ker(T)

I am struggling with an algebra problem here is what we got :  T\begin{bmatrix} a &b \\ c&d \end{bmatrix}= a+d\;\; \; \; and \; \; \;\; L\begin{bmatrix} a &b \\ c&d \end{bmatrix}=\...
13 views

I have the eigenvalue ${\cal{R}} _0$ of $P=\begin{pmatrix}0& \frac{\beta _h}{\mu +\gamma}&\frac{\beta _e}{\kappa (\mu +\gamma)}\\ 0& \frac{\beta _h\gamma}{(\mu +\gamma)(\mu +\alpha)} &\... 0answers 20 views ### The Proximity Operator of a Function with Multiple Affine Mapping Let$f(\mathbf{x}) = g(\mathbf{A}\mathbf{x})$, where$\mathbf{A} \in \mathbb{R}^{M \times N}$is a linear transformation satisfying$\mathbf{A}\mathbf{A}^T = \mathbf{I}$. Then for any$\mathbf{x} \in \...
I want to find the orthogonal projection of the vector $\vec y$ onto a plane. I have $\vec y = (1, -1, 2)$ and a plane that goes through the points \begin{align*}u_1 = (1, 0, 0) \\ u_2 = (1, 1, 1) \\ ...