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CS 229 - Machine Learning
English


Linear Algebra and Calculus refresher
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By Afshine Amidi and Shervine Amidi

General notations

Definitions

Vector ― We note $x\in\mathbb{R}^n$ a vector with $n$ entries, where $x_i\in\mathbb{R}$ is the $i^{th}$ entry:

\[x=\left(\begin{array}{c}x_1\\x_2\\\vdots\\x_n\end{array}\right)\in\mathbb{R}^n\]

Matrix ― We note $A\in\mathbb{R}^{m\times n}$ a matrix with $m$ rows and $n$ columns, where $A_{i,j}\in\mathbb{R}$ is the entry located in the $i^{th}$ row and $j^{th}$ column:

\[A=\left(\begin{array}{ccc}A_{1,1}& \cdots&A_{1,n}\\\vdots&& \vdots\\A_{m,1}& \cdots&A_{m,n}\end{array}\right)\in\mathbb{R}^{m\times n}\]

Remark: the vector $x$ defined above can be viewed as a $n\times1$ matrix and is more particularly called a column-vector.


Main matrices

Identity matrix ― The identity matrix $I\in\mathbb{R}^{n\times n}$ is a square matrix with ones in its diagonal and zero everywhere else:

\[I=\left(\begin{array}{cccc}1&0& \cdots&0\\0& \ddots& \ddots& \vdots\\\vdots& \ddots& \ddots&0\\0& \cdots&0&1\end{array}\right)\]

Remark: for all matrices $A\in\mathbb{R}^{n\times n}$, we have $A\times I=I\times A=A$.


Diagonal matrix ― A diagonal matrix $D\in\mathbb{R}^{n\times n}$ is a square matrix with nonzero values in its diagonal and zero everywhere else:

\[D=\left(\begin{array}{cccc}d_1&0& \cdots&0\\0& \ddots& \ddots& \vdots\\\vdots& \ddots& \ddots&0\\0& \cdots&0&d_n\end{array}\right)\]

Remark: we also note $D$ as $\textrm{diag}(d_1,...,d_n)$.


Matrix operations

Multiplication

Vector-vector ― There are two types of vector-vector products:

- inner product: for $x,y\in\mathbb{R}^n$, we have:

\[\boxed{x^Ty=\sum_{i=1}^nx_iy_i\in\mathbb{R}}\]

- outer product: for $x\in\mathbb{R}^m, y\in\mathbb{R}^n$, we have:

\[\boxed{xy^T=\left(\begin{array}{ccc}x_1y_1& \cdots&x_1y_n\\\vdots&& \vdots\\x_my_1& \cdots&x_my_n\end{array}\right)\in\mathbb{R}^{m\times n}}\]

Matrix-vector ― The product of matrix $A\in\mathbb{R}^{m\times n}$ and vector $x\in\mathbb{R}^{n}$ is a vector of size $\mathbb{R}^{m}$, such that:

\[\boxed{Ax=\left(\begin{array}{c}a_{r,1}^Tx\\\vdots\\a_{r,m}^Tx\end{array}\right)=\sum_{i=1}^na_{c,i}x_{i}\in\mathbb{R}^{m}}\]
where $a_{r,i}^T$ are the vector rows and $a_{c,j}$ are the vector columns of $A$, and $x_i$ are the entries of $x$.


Matrix-matrix ― The product of matrices $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times p}$ is a matrix of size $\mathbb{R}^{n\times p}$, such that:

\[\boxed{AB=\left(\begin{array}{ccc}a_{r,1}^Tb_{c,1}& \cdots&a_{r,1}^Tb_{c,p}\\\vdots&& \vdots\\a_{r,m}^Tb_{c,1}& \cdots&a_{r,m}^Tb_{c,p}\end{array}\right)=\sum_{i=1}^na_{c,i}b_{r,i}^T\in\mathbb{R}^{n\times p}}\]
where $a_{r,i}^T, b_{r,i}^T$ are the vector rows and $a_{c,j}, b_{c,j}$ are the vector columns of $A$ and $B$ respectively.


Other operations

Transpose ― The transpose of a matrix $A\in\mathbb{R}^{m\times n}$, noted $A^T$, is such that its entries are flipped:

\[\boxed{\forall i,j,\quad\quad A_{i,j}^T=A_{j,i}}\]

Remark: for matrices $A,B$, we have $(AB)^T=B^TA^T$.


Inverse ― The inverse of an invertible square matrix $A$ is noted $A^{-1}$ and is the only matrix such that:

\[\boxed{AA^{-1}=A^{-1}A=I}\]

Remark: not all square matrices are invertible. Also, for matrices $A,B$, we have $(AB)^{-1}=B^{-1}A^{-1}$


Trace ― The trace of a square matrix $A$, noted $\textrm{tr}(A)$, is the sum of its diagonal entries:

\[\boxed{\textrm{tr}(A)=\sum_{i=1}^nA_{i,i}}\]

Remark: for matrices $A,B$, we have $\textrm{tr}(A^T)=\textrm{tr}(A)$ and $\textrm{tr}(AB)=\textrm{tr}(BA)$


Determinant ― The determinant of a square matrix $A\in\mathbb{R}^{n\times n}$, noted $|A|$ or $\textrm{det}(A)$ is expressed recursively in terms of $A_{\backslash i, \backslash j}$, which is the matrix A without its $i^{th}$ row and $j^{th}$ column, as follows:

\[\boxed{\textrm{det}(A)=|A|=\sum_{j=1}^n(-1)^{i+j}A_{i,j}|A_{\backslash i,\backslash j}|}\]

Remark: $A$ is invertible if and only if $|A|\neq0$. Also, $|AB|=|A||B|$ and $|A^T|=|A|$.


Matrix properties

Definitions

Symmetric decomposition ― A given matrix $A$ can be expressed in terms of its symmetric and antisymmetric parts as follows:

\[\boxed{A=\underbrace{\frac{A+A^T}{2}}_{\textrm{Symmetric}}+\underbrace{\frac{A-A^T}{2}}_{\textrm{Antisymmetric}}}\]

Norm ― A norm is a function $N:V\longrightarrow[0,+\infty[$ where $V$ is a vector space, and such that for all $x,y\in V$, we have:

- $N(x+y)\leqslant N(x)+N(y)$
- $N(ax)=|a|N(x)$ for $a$ scalar
- if $N(x)=0$, then $x=0$

For $x\in V$, the most commonly used norms are summed up in the table below:

Norm Notation Definition Use case
Manhattan, $L^1$ $||x||_1$ $\displaystyle\sum_{i=1}^n|x_i|$ LASSO regularization
Euclidean, $L^2$ $||x||_2$ $\displaystyle\sqrt{\sum_{i=1}^nx_i^2}$ Ridge regularization
$p$-norm, $L^p$ $||x||_p$ $\displaystyle\left(\sum_{i=1}^nx_i^p\right)^{\frac{1}{p}}$ Hölder inequality
Infinity, $L^{\infty}$ $||x||_{\infty}$ $\underset{i}{\textrm{max }}|x_i|$ Uniform convergence

Linearly dependence ― A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others.

Remark: if no vector can be written this way, then the vectors are said to be linearly independent.


Matrix rank ― The rank of a given matrix $A$ is noted $\textrm{rank}(A)$ and is the dimension of the vector space generated by its columns. This is equivalent to the maximum number of linearly independent columns of $A$.


Positive semi-definite matrix ― A matrix $A\in\mathbb{R}^{n\times n}$ is positive semi-definite (PSD) and is noted $A\succeq 0$ if we have:

\[\boxed{A=A^T}\quad\textrm{ and }\quad\boxed{\forall x\in\mathbb{R}^n,\quad x^TAx\geqslant0}\]

Remark: similarly, a matrix $A$ is said to be positive definite, and is noted $A\succ0$, if it is a PSD matrix which satisfies for all non-zero vector $x$, $x^TAx>0$.


Eigenvalue, eigenvector ― Given a matrix $A\in\mathbb{R}^{n\times n}$, $\lambda$ is said to be an eigenvalue of $A$ if there exists a vector $z\in\mathbb{R}^n\backslash\{0\}$, called eigenvector, such that we have:

\[\boxed{Az=\lambda z}\]

Spectral theorem ― Let $A\in\mathbb{R}^{n\times n}$. If $A$ is symmetric, then $A$ is diagonalizable by a real orthogonal matrix $U\in\mathbb{R}^{n\times n}$. By noting $\Lambda=\textrm{diag}(\lambda_1,...,\lambda_n)$, we have:

\[\boxed{\exists\Lambda\textrm{ diagonal},\quad A=U\Lambda U^T}\]

Singular-value decomposition ― For a given matrix $A$ of dimensions $m\times n$, the singular-value decomposition (SVD) is a factorization technique that guarantees the existence of $U$ $m\times m$ unitary, $\Sigma$ $m\times n$ diagonal and $V$ $n\times n$ unitary matrices, such that:

\[\boxed{A=U\Sigma V^T}\]

Matrix calculus

Gradient ― Let $f:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}$ be a function and $A\in\mathbb{R}^{m\times n}$ be a matrix. The gradient of $f$ with respect to $A$ is a $m\times n$ matrix, noted $\nabla_A f(A)$, such that:

\[\boxed{\Big(\nabla_A f(A)\Big)_{i,j}=\frac{\partial f(A)}{\partial A_{i,j}}}\]

Remark: the gradient of $f$ is only defined when $f$ is a function that returns a scalar.


Hessian ― Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a function and $x\in\mathbb{R}^{n}$ be a vector. The hessian of $f$ with respect to $x$ is a $n\times n$ symmetric matrix, noted $\nabla_x^2 f(x)$, such that:

\[\boxed{\Big(\nabla_x^2 f(x)\Big)_{i,j}=\frac{\partial^2 f(x)}{\partial x_i\partial x_j}}\]

Remark: the hessian of $f$ is only defined when $f$ is a function that returns a scalar.


Gradient operations ― For matrices $A,B,C$, the following gradient properties are worth having in mind:

\[\boxed{\nabla_A\textrm{tr}(AB)=B^T}\quad\quad\boxed{\nabla_{A^T}f(A)=\left(\nabla_Af(A)\right)^T}\] \[\boxed{\nabla_A\textrm{tr}(ABA^TC)=CAB+C^TAB^T}\quad\quad\boxed{\nabla_A|A|=|A|(A^{-1})^T}\]