tutorial on linear algebralinear algebra when is a matrix invertible in general, for an inverse...
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Andrzej Banburski
(based on slides of Xavier Boix, in turn based on thoseof Joe Olson)
Tutorial on Linear AlgebraBrains, Minds & Machines Summer School 2018
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
Convolution as Toeplitz matrix
For experts: even a convolution operation can be recast as matrix multiplication:
https://stackoverflow.com/questions/48768555/toeplitz-matrix-with-an-image
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Linear Algebra
Norms
A function that measures the “size” of a vector is called a norm.
The 𝐿𝑝 norm us given by
More generally, the norm has to satisfy the following:• 𝑓 𝑥 = 0 ⇒ 𝑥 = 0• 𝑓 𝑥 + 𝑦 ≤ 𝑓 𝑥 + 𝑓(𝑦) (triangle inequality• ∀𝛼 ∈ ℝ, 𝑓 𝛼𝑥 = 𝛼 𝑓 𝑥
𝑥 𝑝 =
𝑖
𝑥𝑖𝑝
1𝑝
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Linear Algebra
Norms
The most commonly used norm for vectors is 𝑝 = 2, which is compatible with the inner product
Another useful norm is the 𝐿∞ norm, also known as max norm:
The most natural measure of matrix “size” is the Frobenius norm:
𝑥 2 = 𝑥 ⋅ 𝑥
𝑥 ∞ = max𝑖
𝑥𝑖
𝐴 𝐹 =
𝑖,𝑗
𝐴𝑖𝑗2
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Linear Algebra
Trace
The trace of a matrix is the sum of its’ diagonal entries
Some useful properties:
• 𝑇𝑟 𝐴 = 𝑇𝑟 𝐴𝑇
• 𝑇𝑟 𝐴𝐵𝐶 = 𝑇𝑟 𝐶𝐴𝐵 = 𝑇𝑟(𝐵𝐶𝐴) (if defined)
𝑇𝑟 𝐴 =
𝑖
𝐴𝑖𝑖
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Linear Algebra
Determinant
The determinant is a value that can be computed for a square matrix.
For a 2x2 matrix it is given by 𝑎 𝑏𝑐 𝑑
= 𝑎𝑑 − 𝑏𝑐.
Interpretation: volume of parallelepiped is the absolute value of the determinant of a matrix formed of row vectors r1, r2, r3.
In general for an (n,n) matrix it is given by
Computed over all permutations 𝜎 of the set {1,…,n}.
det 𝐴 =
𝜎∈𝑆𝑛
sgn(𝜎)ෑ
𝑖=1
𝑛
𝐴𝑖,𝜎(𝑖)
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
When is a matrix invertible
In general, for an inverse matrix 𝐴−1 to exist, 𝐴 has to be square and its’ columns have to form a linearly independent set of vectors – no column can be a linear combination of the others.
A necessary and sufficient condition is that det 𝐴 ≠ 0.
Finding the inverse is usually quite arduous, even though an explicit expression exists:
𝐴−1 =1
det(𝐴)
𝑠=0
𝑛−1
𝐴𝑠
𝑘1,𝑘2,…,𝑘𝑛
ෑ
𝑙=1
𝑛−1(−1)𝑘𝑙+1
𝑘𝑙! 𝑙𝑘𝑙
𝑇𝑟 𝐴𝑙𝑘𝑙
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
Eigendecomposition and SVD
In fact, if a square matrix has n linearly independent eigenvectors, it can always be diagonalized 𝐴 = 𝑈 𝑑𝑖𝑎𝑔 𝜆 𝑈𝑇.
In this case we also immediately get the inverse matrix
For a non-square 𝑚 × 𝑛 matrix we can at best perform the Singular Value Decomposition: 𝐴 = 𝑈 𝐷 𝑉𝑇, where 𝑈 is an 𝑚 ×𝑚 orthogonal matrix, 𝐷 is diagonal 𝑚 × 𝑛 and 𝑉 another n× 𝑛 orthogonal matrix. Elements of 𝐷 are known as singular values
𝐴−1 = 𝑈 𝑑𝑖𝑎𝑔1
𝜆𝑈𝑇
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Linear Algebra
Moore-Penrose Pseudoinverse
Matrix inversion is not defined for non-square matrices. Suppose we have a linear equation 𝐴𝑥 = 𝑦 and we want to solve for 𝑥. • If 𝐴 is taller than wider, there might be no solutions• If it is wider than taller, there might be many solutions.
The pseudoinverse is defined as
In practice we calculate it as 𝐴+ = 𝑉𝐷+𝑈𝑇, where 𝑈,𝐷, 𝑉 are the SVD of 𝐴and 𝐷+ is calculated by taking the reciprocal of non-zero singular values and taking the transpose of the result. (Note this is clearly discontinuous)
𝐴+ = lim𝛼→0+
𝐴𝑇𝐴 + 𝛼𝐼−1𝐴𝑇
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Linear Algebra
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Thanks!