**Linear Algebra and Calculus translation** [[webpage]](https://stanford.edu/~shervine/teaching/cs-229/refresher-algebra-calculus)

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**1. Linear Algebra and Calculus refresher**

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**2. General notations**

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**3. Definitions**

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**4. Vector ― We note x∈Rn a vector with n entries, where xi∈R is the ith entry:**

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**5. Matrix ― We note A∈Rm×n a matrix with m rows and n columns, where Ai,j∈R is the entry located in the ith row and jth column:**

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**6. Remark: the vector x defined above can be viewed as a n×1 matrix and is more particularly called a column-vector.**

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**7. Main matrices**

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**8. Identity matrix ― The identity matrix I∈Rn×n is a square matrix with ones in its diagonal and zero everywhere else:**

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**9. Remark: for all matrices A∈Rn×n, we have A×I=I×A=A.**

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**10. Diagonal matrix ― A diagonal matrix D∈Rn×n is a square matrix with nonzero values in its diagonal and zero everywhere else:**

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**11. Remark: we also note D as diag(d1,...,dn).**

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**12. Matrix operations**

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**13. Multiplication**

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**14. Vector-vector ― There are two types of vector-vector products:**

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**15. inner product: for x,y∈Rn, we have:**

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**16. outer product: for x∈Rm,y∈Rn, we have:**

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**17. Matrix-vector ― The product of matrix A∈Rm×n and vector x∈Rn is a vector of size Rn, such that:**

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**18. where aTr,i are the vector rows and ac,j are the vector columns of A, and xi are the entries of x.**

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**19. Matrix-matrix ― The product of matrices A∈Rm×n and B∈Rn×p is a matrix of size Rn×p, such that:**

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**20. where aTr,i,bTr,i are the vector rows and ac,j,bc,j are the vector columns of A and B respectively**

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**21. Other operations**

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**22. Transpose ― The transpose of a matrix A∈Rm×n, noted AT, is such that its entries are flipped:**

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**23. Remark: for matrices A,B, we have (AB)T=BTAT**

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**24. Inverse ― The inverse of an invertible square matrix A is noted A−1 and is the only matrix such that:**

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**25. Remark: not all square matrices are invertible. Also, for matrices A,B, we have (AB)−1=B−1A−1**

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**26. Trace ― The trace of a square matrix A, noted tr(A), is the sum of its diagonal entries:**

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**27. Remark: for matrices A,B, we have tr(AT)=tr(A) and tr(AB)=tr(BA)**

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**28. Determinant ― The determinant of a square matrix A∈Rn×n, noted |A| or det(A) is expressed recursively in terms of A∖i,∖j, which is the matrix A without its ith row and jth column, as follows:**

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**29. Remark: A is invertible if and only if |A|≠0. Also, |AB|=|A||B| and |AT|=|A|.**

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**30. Matrix properties**

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**31. Definitions**

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

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**33. [Symmetric, Antisymmetric]**

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**34. Norm ― A norm is a function N:V⟶[0,+∞[ where V is a vector space, and such that for all x,y∈V, we have:**

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**35. N(ax)=|a|N(x) for a scalar**

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**36. if N(x)=0, then x=0**

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**37. For x∈V, the most commonly used norms are summed up in the table below:**

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**38. [Norm, Notation, Definition, Use case]**

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**39. 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.**

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

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**41. Matrix rank ― The rank of a given matrix A is noted 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.**

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**42. Positive semi-definite matrix ― A matrix A∈Rn×n is positive semi-definite (PSD) and is noted A⪰0 if we have:**

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**43. Remark: similarly, a matrix A is said to be positive definite, and is noted A≻0, if it is a PSD matrix which satisfies for all non-zero vector x, xTAx>0.**

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**44. Eigenvalue, eigenvector ― Given a matrix A∈Rn×n, λ is said to be an eigenvalue of A if there exists a vector z∈Rn∖{0}, called eigenvector, such that we have:**

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**45. Spectral theorem ― Let A∈Rn×n. If A is symmetric, then A is diagonalizable by a real orthogonal matrix U∈Rn×n. By noting Λ=diag(λ1,...,λn), we have:**

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**46. diagonal**

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**47. Singular-value decomposition ― For a given matrix A of dimensions m×n, the singular-value decomposition (SVD) is a factorization technique that guarantees the existence of U m×m unitary, Σ m×n diagonal and V n×n unitary matrices, such that:**

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**48. Matrix calculus**

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**49. Gradient ― Let f:Rm×n→R be a function and A∈Rm×n be a matrix. The gradient of f with respect to A is a m×n matrix, noted ∇Af(A), such that:**

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

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**51. Hessian ― Let f:Rn→R be a function and x∈Rn be a vector. The hessian of f with respect to x is a n×n symmetric matrix, noted ∇2xf(x), such that:**

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

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

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**54. [General notations, Definitions, Main matrices]**

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**55. [Matrix operations, Multiplication, Other operations]**

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**56. [Matrix properties, Norm, Eigenvalue/Eigenvector, Singular-value decomposition]**

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**57. [Matrix calculus, Gradient, Hessian, Operations]**

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