# Questions tagged [trace]

For questions about trace, which can concern matrices, operators or functions. If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.

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### Trace of matrix product involving identity and powers

Suppose we know $\text{Tr}(A)=a$. Is there a closed formula for obtaining \begin{eqnarray} \text{Tr}(A(A-I)^n), \end{eqnarray} for any $n=1,2,...$ and with $I$ being the identity matrix? Such products ...
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### Is it true that $tr(APA^T)$ > $tr(AQA^T)$, if $tr(P)$ > $tr(Q)$

Assuming $P$ and $Q$ and positive definite matrix. Is it true that $tr(APA^T)$ > $tr(AQA^T)$, when $tr(P)$ > $tr(Q)$. [EDIT after first answer: not just that, actually $P_{ii} > Q_{ii}$ for ...
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### Proof of Conjecture on `Block-Orthogonalisation Always Reduces Trace'

I have a symmetric positive definite matrix, ${\bf D}\in\mathbb{R}^{NK\times NK}$, which is made up of $K$, $N\times N$, blocks: {\bf D} = \begin{bmatrix}{\bf D_{11}} & {\bf D_{12}...
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### Trace of Operators [closed]

Let $H_1$ and $H_2$ be two Hilbert Spaces. Let A be an bounded linear operator between $H_1$ and $H_2$ such that $AA^*$ is traceclass, where $A^*$ denotes the adjoint Operator. Is it true that we have ...
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### trace of a matrix squared:a formula

Can someone explain to me why $$\operatorname{Tr}K^2=\sum_{\alpha,\beta} K_{\alpha\beta}K^{\alpha\beta}$$
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### Minimizing the trace of matrix using matrix calculus

If I have a function $f(X)$ that returns a square matrix, and $f(X)$ is a convex function with respect to matrix $X$, I sometimes see literature just tell me $\partial f(X)/ \partial X = 0$ will ...
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### A Trace inequality

For $A,B$ positive self-adjoint matrices do we have $$\operatorname{tr}(AB^2A) \leq \|A\|\|B\|\operatorname{tr}(BA),$$ where $\|\cdot\|$ denotes the operator norm, and $\operatorname{tr}$ denotes the ...
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### If the trace of a matrix equals its rank, is it idempotent?

It is well-known and can easily be proven that if a matrix $A$ is idempotent, then its trace equals its rank: $$A^2 = A \Rightarrow \mathrm{tr}(A) = \mathrm{rk}(A)$$ Does the inverse also hold? If ...
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### Does this matrix exist? A commonly encountered puzzle

Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of ...
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### Symmetric and skew-symmetric matrices [closed]

Is the product of a symmetric and skew-symmetric matrix symmetric? does this answer remain accurate if the skew-symmetric matrix is replaced by a non-symmetric matrix? If not, what happens then?
1 vote
I am new to linear algebra; someone on Quora posted this "shortcut" method to find the characteristic equation of a $3\times 3$ matrix. Though they demonstrated it through an example, here's ...