# Questions tagged [transpose]

In linear algebra, the transpose of a matrix is another matrix whose i-th row and j-th column is the j-th row and i-th column of the original matrix.

533 questions
Filter by
Sorted by
Tagged with
46 views

### How to scale a fat matrix to ensure its rows are orthonormal?

I have the following matrix $${\bf B} = \begin{bmatrix} 0&-4.2423&4.2423&1.4871\\ 1.6532&-1.2735& -1.2735&0.0024\\ 0 & -0.2805 & 0.2805 & -0.8823 \end{bmatrix}$$ I ...
12 views

### Finding the conjugate/hermitian transpose of a transformation

Hopefully, I'm using the correct terms/names of things, mainly because the language in which I study is not English. Given the operator $T$, in this case is the derivative operator , with the inner ...
• 115
1 vote
72 views

### Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

In a linear algebra textbook, I was given the following problem: If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric. I know that there are relatively ...
• 269
29 views

### Matrix of adjoint linear map is transposed matrix

I am stuck with some exercise and hope that someone can give me a hint: Let $V$ and $W$ be vector spaces and let $B_V = \big\{ v_1, \ldots, v_n \big\}$ and $B_w = \big\{ w_1, \ldots, w_n \big\}$ be ...
• 13
1 vote
38 views

### When is a sum of products of two matrices and the transposes positive definite?

Let $X$ and $Y$ be $n \times m$ matrices. The matrix $$A = X^TY + Y^TX$$ will be a $m \times m$ square, symmetric matrix. Is it possible to say: i) when is $A$ is positive definite? and when it is ...
• 45
1 vote
41 views

### Are left and right operator norms equal?

If $V$ is a finite dimensional normed vector space and I identify row and column vectors then I can define the action of a matrix both from the left and from the right. I can define the left and ...
47 views

### Does $A=B$ imply that the rank of $A$ equals the rank of $B$?

I am trying to understand a poof of the clubs of oddtown theorem, and I am stuck at this step: If $A$ is an $m \times n$ matrix and $AA^T=I_m$, then the rank of $AA^T$ is AT LEAST $m$. Now if $A=B$ ...
• 2,147
84 views

48 views

• 5,070
1 vote
37 views

### Is the transpose of $T$ also the dual map of $T$?

Suppose I have $T: V \to W$. Then $T$ defines a map $T': W' \to V'$ that some sources (e.g. Axler) call the dual map of $T$. And $T$ defines a map $T^t: W' \to V'$ that some sources (e.g. Hoffman and ...
• 2,077
45 views

### Simple Matrix Equation.

$$(2\begin{bmatrix}2 &1 \\-1 &3\end{bmatrix} - 5A^{-1})^{T} = (4A^{T})^{-1}$$ I have approached this question by inverting the transpose and inverse operation on the LHS and then distributing ...
23 views

### Meaning of left multiplying a matrix A with another B and then right multiply the result with a transposed B?

I am currently in the process of learning Kalman Filters and facing the following Equations: $$\vec x_k = F\vec x_{k-1} + B\vec u_k$$ $$P_k = FP_{k-1}F^T + Q$$ $\vec x_k$: state vector. $P_k$: ...
21 views

### Confusion on allowed operation on row vectors

Assume we are given the following scalar quantity: \begin{align} q &= v^T A w \\ & =(v_1 \,\, v_2)\begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{pmatrix}\begin{pmatrix} w_1\\ ...
• 2,119
51 views

• 741
69 views

### Is it possible for $AA^T$ to be a nilpotent matrix if neither $A$ nor $A^T$ are?

If $A$ is a square, non-nilpotent matrix with real-valued elements, and its transpose is $A^T$, then is it ever possible for $AA^T$ to be nilpotent? What if we allow complex-valued elements? Is $AA^H$ ...
• 1,137
31 views

### Symmetric matrices, transpose matrices [duplicate]

i have a problem from my linear algebra exam that i couldn't solve: $B$ is an element of $M_n(\mathbb{R})$ and we now that $B\cdot B=B$ and that $B^T \cdot B=B \cdot B^T$, we now have to show that ...
• 17
1 vote
32 views

### linear algebra - linear transformation and transpose

Every linear transformation can be represented as a matrix. If I take a matrix representation of a certain linear transformation and transpose it, what kind of operator do I get? when does the ...
• 11
34 views

### Common factor in sum of transpose matrices

I have the following linear matrix inequality: $(B K_1)^T + BK_1 < -2A$ where B is 2x1, $K_1$ is 1x2 and A is 2x2. Is it possible to find $K_1$ as a common factor in the left-hand side of the ...
1 vote
63 views

### Please tell me a simple proof for the following equality: $\operatorname{rank} A=\operatorname{rank} A A^T.$

Please tell me a simple proof for the following equality: $$\operatorname{rank} A=\operatorname{rank} A A^T.$$ I proved the above equality as follows: I found the following formula in "Linear ...
• 6,159
62 views

### Why do we define Symplectic groups with Transpose and not hermitian adjoint?

In the definition of a symplectic group over a field, we take the definition: all matrices $S$ of some dimension $2n$, such that $$S^T \Omega S = \Omega$$ where omega is a skew-symmetric matrix (bi-...
• 434
1 vote
40 views

### What is the most common notation for a transpose of a gradient of a function?

Say we have a function where the argument is pretty large horizontally, like $f(x^k,\lambda_k, \mu_k)$, or something. Then you take the transpose of this gradient for whatever reason, be it for a ...
• 929
28 views

### Hat matrix: various results in linear algebra.

In the question, the hat matrix $H = I-x(x^Tx)^{-1}x^T$ is an $n$ x $n$ matrix. $x$ is a $n$ x $p$ vector, whose columns are linearly dependent. How do I show that $Hx = 0$, $H^T = H$, and $H^2 = H$. ...
• 307
66 views

### Positive definiteness [duplicate]

Let $A$ be a positive definite matrix is it true that also $A+A^T$ is positive definite? If it is true, how to prove it? I try to employ the definition using eigenvalues but i'm not able to proceed. ...
• 316
30 views

### What happens to the dependance of the columns of a matrix once manipulated?

Lets say that we have a square matrix ( n x n ), and Ax=B has only one solution (where B ∈ 𝑅 ). We can tell from the fact that Ax=B has only one solution that the columns of A will be linearly ...
• 39
33 views

### Geometric sum of transpose matrices

Is there any mapping between the geometric sum of $A$ and the geometric sum of $A^T$? To be precise are there conditions under which $(I-A)^{-1}$ is approximately close to $(I-A^T)^{-1}$, other than ...
59 views

### Consider the following matrices. Calculate the results of the operations below, where the operations are allowed. [duplicate]

I have added a image of the question below, but to explain: suppose you have two matrices, one called C and another called B, how do you work out (CB)^T? is it a matter of multiplying C and B and then ...
1 vote
117 views

• 27
73 views

### Is $\operatorname{rank}(A) = \operatorname{rank}( A^T )$ always true?

The answer key for my exam says that it's not always true, but I can't seem to come up with an explanation for this or find a matrix that makes this equation untrue. Thanks
• 31
56 views

### Why is the orthogonal complement of col(A) = ker(A*)?

col(A) = column space of matrix A ker(A*) = kernel (nullspace) of matrix A* (where A* is the conjugate transpose which is just AT for real matrices.) Why does orthogonal complement col(A) = ker(A*) ? ...
• 11
105 views

### Which change of basis matrix makes a companion matrix similar to its transpose?

I know that a companion matrix is similar to its transpose e.g by Smith normal form. When the characteristic polynomial splits, companion matrices become Jordan blocks. A Jordan block is similar to ...
• 12.7k
1 vote
131 views

### The inverse of a permutation matrix is its transpose

I do this exercise but I don't know if this okay. $$P^{-1}_{\sigma}=P^{T}_{\sigma}=P_{\sigma^{-1}}$$ Where $P_{\sigma}=[e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]$ The first thing that I do is ...
• 121
If you define the inner product $\langle u,v\rangle=u^Tv$, then the fact that $\langle Au,v\rangle=\langle u,A^T\rangle$ where $A$ is a matrix defined over $\Bbb{R}$ is easily seen. Likewise, if you ...