# Questions tagged [matrix-calculus]

Matrix calculus is about doing calculus, especially derivative and infinite series over spaces of vectors and matrices.

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### Gradient of Frobenius inner product involving a homogeneous function

Let $f:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{m \times n}$ be a differential function. I want to take the gradient of the Frobenius inner product between $X$ and $f(X)$ with respect to $X$. ...
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### Closed form for summation of matrix multiplication by its transpose

Assume $A_{N\times{N}}$ is a square matrix. Is there a closed form for $$I+A+AA'+(AA')A'+((AA')A')A'+...$$ Essentially a closed form for multiplication of the matrix by its transpose for infinite ...
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### Is there any formula for the inverse of a 3x3 matrix concerning its row vectors?

I have a 3x3 matrix $A$: $$A = \begin{bmatrix} \vec a_0^T \\ \vec a_1^T \\ \vec a_2^T \end{bmatrix} \in \mathbb R^{3\times 3},$$ where $\vec a_i \in \mathbb R^3$. Is there any formula of $A^{-1}$ ...
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### Proving the equivalence of two notions of ODE for matrices

Let $\mathcal{H}$ be a complex finite-dimensional Hilbert space with basis $\mathcal{B} = \{e_{1},...,e_{n}\}$. Suppose that, for each $t \in \mathbb{R}$, $M(t)$ is a linear operator on $\mathcal{H}$. ...
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### Differentiate the summation over an index of a rank 3 tensor.

$X^{i,j,\lambda}$ is a rank 3 tensor, $W^{i,j}$ is a matrix. Is it possible to analytically solve the following? $\frac{d}{dX}\left|| W - \sum_\lambda X \right||^{2}$ I know with matrices we can ...
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### What the rank of a matrix with the elements of one column to be infinity? [closed]

Suppose the $m\times m$ real matrix $A$ is positive definite, it is without doubt that $\mathrm{rank} (A) = m$. Now, if all the elements in the first column and first row are assigned to be infinity, ...
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### Covariance matrix of Linear Transformation

I need help. I'm looking at matrix-vector multiplication. Both vector (x) and matrix (A) are random and independent. The vector has a mean and a covariance matrix. And the matrix has the mean and ...
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### Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$

Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$ with $\mathbf{A}\in\mathbb{R}^{M\times N}$ and $\mathbf{X}\in\mathbb{R}^{N\times D}$, and ...
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### Derivative of transpose of matrix

Let $X$ be a $n*n$ matrix with its entry indices increasing along every column.For example,when $n = 2$, $X =\left( \begin{matrix}x_1 & x_3\\ x_2 & x_4\end{matrix} \right)$.Let $\rm{vec}(X)$ ...
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### Differentiating a matrix with respect to a vector

In multivariate linear model, I have come across the following matrix-valued function of $\beta \in \Bbb R^p$. $$\beta \mapsto(y-X\beta)(y-X\beta)^{T}$$ where matrix $X \in \Bbb R^{n \times p}$ and ...
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### Power series for $(A+\epsilon I)^{-1}$ for large $\epsilon$

I am trying to figure out a power series for $(A+\epsilon I)^{-1}$ when $A$ is an invertible matrix and $\epsilon$ is large. The Neumann series can be used when $\epsilon$ is small. Is there something ...
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### ODE for the inverse of fundamental solution

Let $A:[0,T]\to \mathbb{R}^{n\times n}$ be a continuous function, and let $\Phi$ satisfy $\frac{d}{d t} \Phi_t=A_t \Phi_t$ for all $t\in [0,T]$, and $\Phi_0=I_n$. I hope to prove that the solution to ...
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I just started learning matrix derivatives and I'm trying to find derivative of $$f(T) = |I-2iT\Sigma|^{-\frac{n}{2}},$$ where $T$ and $\Sigma$ are symmetric matrices. This is what I already have: \... • 105 0 votes 0 answers 45 views ### Analatic functions with two variables applied to square matrices I understand if square matrix A=P_{1}^{-1}D_{1}P_{1}, where D_{1} is a diagonal matrix, we can write f(A) as P_{1}^{-1}f(D_{1})P_{1}. My question is how to define f(A,B) where A=P_{1}^{-1}... • 53 0 votes 2 answers 48 views ### Derivative of the product of a matrix scalar function and a matrix with respect to a matrix How to take derivative of \begin{align*} \dfrac{\partial\left[\operatorname{tr}(\boldsymbol{A}^2)\cdot\boldsymbol{A}^3\right]}{\partial\boldsymbol{A}}=\,? \end{align*} with respect to a matrix \... • 2,861 3 votes 1 answer 112 views ### How to find the coefficients of the second eigenvector? I have a 2\times 2 real symmetric matrix:\begin{pmatrix} A & C \\ C & B \end{pmatrix} $$and I know that the eigenvalues are:$$\lambda_{\pm} = \frac{1}{2}(A+B)\pm \frac{1}{2}\sqrt{(A-B)^...
Suppose there is a 2$\times$2 matrix $M=\matrix[a\ ,b\ ;\ c\ ,d]$, which $$M \matrix[y_1\ ,\ y_2]^T=\matrix[y_1'\ ,\ y_2']^T$$ if the elements of the matrix M are known, is it possible to calculate ...
I want to find the derivative of $x^*M(x)x$ with respect to $x$. $x \in R^2$ is a vector and $M(x)$ is a smooth symmetric matrix function that maps $R^2 \to R^{2\times 2}$. I can write the matrix as ...