Linear Algebra Technological Applications

Linear Algebra Technological Applications Edmar Candeia Gurjão [email protected] Universidade Federal de Campina Grande

November 2017

Edmar Candeia Gurjão (UFCG/CEEI/DEE)

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Introdução

Edmar Candeia Gurjão

Department of Electrical Engineering - UFCG; Interests: Digital Signal Processing, Compressed Sensing;


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Introdução

Edmar Candeia Gurjão

Analog domain: Calculus, Analog Computer. Digital domains: Linear Algebra. Digital Computer.


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Introdução

Cursos de Álgebra Linear

• Subjects related to Linear Algebra MatriX Determinants Sistems of LInear Equations Vector Spaces Eigenvalues and eigenvectors Diagonalization SVD, LU, and other decompositions Principal Component Analysis (PCA) Independent Component Analysis (ICA) ....


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Linear Algebra

Definition × Application

• Matrix Definition A = [aij ], aij ∈ R 1 ≤ i ≤ M, 1 ≤ j ≤ N

And? Ex.:

1 2 3   A= 4 5 6  7 8 9 



Matrix multiplication: simple and easyl! https://docs.gimp.org/en/plug-in-convmatrix.html


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Linear Algebra


• Determinants Definition Given a swuare matrix AM×M its determinant is given by det(A) =

M X

aij δij ,

j=1

where ∆ij is the cofactor given by ∆ij = (−1)i+j det(A)ij , and det(A)ij is the determinant of the matrix obtained from A by removing row i and column j. And? Volume of a solid..


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Linear Algebra


• Inverse matrix Definition Given a matrix A, its inverse is a matrix A−1 such that A.A−1 = I, where I is the identity matrix. And? Criptography. Inverse propagation media.


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Linear Algebra


• Basis of a Vector Space Definition A set of vectors {v1 , v2 , ..., vn } is a basis for a space vector V if: A Linear Independent (LI) set; A set that geneartes the space V ; Obeys a set of axioms. And? Compact representation of a set of vectors.


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