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matrices - intro

Week 1 has basics of vectors, matrix form of vecs, and determinants of those vectors

Why​

  1. Why is vector and matrix important
  2. Vec <-> matrix
  3. What's determinant? Why is that useful? How to calculate.
  1. Determinants are mathematical expressions derived from square matrices that hold significant importance in various areas of linear algebra and beyond. Here are the key reasons why determinants are important and their applications:

Importance of Determinants:​

  1. Matrix Inversion:

    • Determinants are used to determine whether a square matrix has an inverse. A matrix is invertible (non-singular) if and only if its determinant is non-zero.

  2. Solving Linear Systems:

    • Determinants play a crucial role in solving systems of linear equations. The solution can be found using Cramer's rule, which expresses the solution in terms of determinants.

  3. Eigenvalues and Eigenvectors:

    • Determinants are involved in finding the eigenvalues of a matrix, as the characteristic polynomial (used to find eigenvalues) is derived from the determinant of (A - \lambda I).

  4. Volume Calculation:

    • In geometry, the determinant of a matrix composed of vectors can be used to calculate the volume of parallelepipeds and other geometric shapes.

  5. Transformation Properties:

    • Determinants describe how linear transformations scale volumes. For example, the determinant of a transformation matrix indicates the scaling factor of the transformation.

  6. Area and Volume Interpretations:

    • Determinants provide the area and volume interpretation for 2 x 2 and 3 x 3 matrices respectively, relating to the signed area and volume of parallelograms and parallelepipeds formed by row or column vectors.

  7. Stability and Control Theory:

    • In control theory, determinants are used in analyzing the stability of systems. The Routh-Hurwitz criterion, which involves determinants, is used to determine the stability of a linear system.

  8. Differential Equations:

    • Determinants are used in the context of differential equations, particularly in the Wronskian determinant to test the linear independence of solutions.

Applications of Determinants:​

  1. Cramer's Rule:

    • Solving linear equations (AX = B) using determinants. If (A) is an (n \times n) matrix, the solutions (X_i) are given by (X_i = \frac{\det (A_i)}{\det (A)}), where (A_i) is the matrix formed by replacing the (i)-th column of (A) with (B).

  2. Characteristic Polynomial:

    • Finding eigenvalues (\lambda) of a matrix (A) by solving (\det (A - \lambda I) = 0).

  3. Geometric Interpretations:

    • Computing areas and volumes: For example, the area of a triangle in 2 D space can be found using the determinant of a matrix formed by its vertices.

  4. Transformation Matrices in Graphics:

    • Determinants are used in computer graphics to perform transformations like rotation, scaling, and shearing, ensuring that the transformations are invertible.

  5. Cryptography:

    • Determinants are used in certain encryption algorithms, such as Hill cipher, where the invertibility of the matrix (determinant is non-zero) is crucial.

  6. Quantum Mechanics:

    • In quantum mechanics, determinants (specifically, the determinant of a matrix known as the Slater determinant) are used to describe the wavefunctions of multi-electron systems.

  7. Economics and Game Theory:

    • Determinants are used in input-output analysis in economics to determine the viability and stability of economic systems. In game theory, determinants are used in the analysis of equilibrium points.

  8. Engineering:

    • In civil and mechanical engineering, determinants are used to solve for forces and moments in statically determinate structures.

Summary:​

Determinants are essential in mathematics due to their ability to succinctly encapsulate properties of matrices and their applications span across solving linear equations, determining matrix invertibility, finding eigenvalues, and various practical applications in geometry, engineering,

More :​

  • Determinants help in determining the unique solution of a system of linear equations. This unique solution exists when the matrix representing the coefficients of the system is square and its determinant is non-zero.

  • Square is impotatnt as, say we need to solve for x, z, y, a now we cannot do that with 3 eqs, more is fine? So less in one direction is incomplete, not less in other, Vice versa same for more

  • The inverse of a matrix A is another matrix, denoted as inverse, such that when A is multiplied by inverse, the result is the identity matrix III. The identity matrix is analogous to the number 1 in multiplication of real numbers; it leaves any vector unchanged when multiplied by it. Mathematically: $A.A^{-1}=A^{-1}.A=I$

  • Think of a matrix as a function that transforms vectors in space. For instance, a 2x2 matrix might rotate, stretch, or shear a vector in 2D space. The inverse matrix undoes this transformation. If matrix A stretches a vector, A inverse compresses it back to its original form. If A rotates a vector, inverse rotates it back.

  • Consider transformations in a coordinate system. A square matrix might transform a unit square into a parallelogram. The determinant of this matrix gives the area of the parallelogram. The inverse matrix would transform this parallelogram back into the unit square, essentially reversing the transformation.

  • A square matrix AAA has an inverse if and only if its determinant is non-zero. This condition ensures that the transformation represented by AAA is not "degenerate" (i.e., it doesn't collapse the space into a lower dimension where information is lost).