# Linear Algebra II: Matrix Theory

## Video Lectures

Displaying all 35 video lectures.

Lecture 1Play Video |
Matrix Inverse over the Complex NumbersMatrix Theory: Find the solution to the following system of linear equations over C using the inverse of a 2 x 2 matrix: (1+i)x + 2iy = 2 -ix + y = 2i |

Lecture 2Play Video |
Cramer's Rule over the Complex NumbersMatrix Theory: Using Cramer's Rule, find the solution to the following system of linear equations over the complex numbers: 2i x1 + 3 x2 = 2; (-1-i) x1 + x2 - x3 = 0; -3 x1 + i x3 = i |

Lecture 3Play Video |
Gaussian Elimination over Z/3Matrix Theory: Find all solutions to the system of linear equations over the finite field Z/3: x+ 2y = 1, 2x + y = 2. |

Lecture 4Play Video |
Matrix Inverse over Z/7Matrix Theory: Solve the following system of equations over Z/7 by finding A^{-1}: 3x + 2y = 4, x + 5y = 1. |

Lecture 5Play Video |
Cramer's Rule over Z/5Matrix Theory: Using Cramer's Rule, we solve a system of linear equations Ax = b over the finite field Z/5. |

Lecture 6Play Video |
Example of Skew-Symmetric MatrixMatrix Theory: Let a be an invertible skew-symmetric matrix of size n. Show that n is even, and then show that A^{-1} is also skew-symmetric. We show the identities (AB)^T = B^T A^T and (AB)^{-1} = B^{-1}A^{-1}. |

Lecture 7Play Video |
Example of Simultaneous DiagonalizationMatrix Theory: Find a joint eigenbasis for the commuting matrices A = [2 2 \ 2 2] and B = [1 2 \ 2 1]. That is, find a basis of eigenvectors that simultaneously diagonalize A and B. |

Lecture 8Play Video |
Positive Semi-Definite Matrix 1: Square RootMatrix Theory: Let A be an nxn matrix with complex entries. Assume that A is (Hermitian) positive semi-definite. We show that A has a unique (Hermitian) positive definite square root; that is, a PSD matrix S such that S^2 = A. The key ingredient is the Spectral Theorem for C^n. Example in Part 2. |

Lecture 9Play Video |
Positive Semi-Definite Matrix 2: Spectral TheoremMatrix Theory: Following Part 1, we note the recipe for constructing a (Hermitian) PSD matrix and provide a concrete example of the PSD square root. We prove the Spectral Theorem for C^n in the remaining 9 minutes. |

Lecture 10Play Video |
Positive Semi-Definite Matrix 3: Factorization of Invertible MatricesMatrix Theory: Let A be an invertible nxn matrix with complex entries. Using the square root result from Part 1, we show that A factors uniquely as PX, where P is unitary and X is (Hermitian) positive definite. |

Lecture 11Play Video |
Cayley-Hamilton Theorem for 2x2 MatricesMatrix Theory: We state and prove the Cayley-Hamilton Theorem for a 2x2 matrix A. As an application, we show that any polynomial in A can be represented as linear combination of A and the identity matrix I. |

Lecture 12Play Video |
Inverse of a Matrix Using the Cayley-Hamilton TheoremMatrix Theory: Suppose a 2 x 2 real matrix has characteristic polynomial p(t) = t^2 - 2t + 1. Find a formula for A^{-1} in terms of A and I. Verify the formula for A = [1 2\ 0 1]. |

Lecture 13Play Video |
Cayley-Hamilton Theorem: General CaseMatrix Theory: We state and prove the Cayley-Hamilton Theorem over a general field F. That is, we show each square matrix with entries in F satisfies its characteristic polynomial. We consider the special cases of diagonal and companion matrices before giving the proof. |

Lecture 14Play Video |
Cayley-Hamilton Theorem: Example 1Matrix Theory: We verify the Cayley-Hamilton Theorem for the real 3x3 matrix A = [ / / ]. Then we use CHT to find the inverse of A^2 + I. |

Lecture 15Play Video |
Cayley-Hamilton Theorem Example 2Matrix Theory: Let A be the 3x3 matrix A = [1 2 2 / 2 0 1 / 1 3 4] with entries in the field Z/5. We verify the Cayley-Hamilton Theorem for A and compute the inverse of I + A using a geometric power series. |

Lecture 16Play Video |
Example of Invariant SubspaceMatrix Theory: Let T: R^4 to R^4 be the linear transformation that sends v to Av where A = [0 0 0 -1 \ 1 0 0 0 \ 0 1 0 -2 \ 0 0 1 0]. Find all subspaces invariant under T. |

Lecture 17Play Video |
Overview of Jordan Canonical FormMatrix Theory: We give an overview of the construction of Jordan canonical form for an nxn matrix A. The main step is the choice of basis that yields JCF. An example is given with two distinct eigenvalues. |

Lecture 18Play Video |
Example of Jordan Canonical Form: 2x2 MatrixMatrix Theory: Find the Jordan form for the real 2 x 2 matrix A = [0 -4 \ 1 4]. For this matrix, there is no basis of eigenvectors, so it is not similar to a diagonal matrix. One alternative is to use Jordan canonical form. |

Lecture 19Play Video |
Example of Jordan Canonical Form: General PropertiesMatrix Theory: A real 8x8 matrix A has minimal polynomial m(x) = (x-2)^4, and the eigenspace for eigenvalue 2 has dimension 3. Find all possible Jordan Canonical Forms for A. |

Lecture 20Play Video |
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1Matrix Theory: Find a matrix P that puts the real 4x4 matrix A = [2 0 0 0 \ 0 2 1 0 \ 0 0 2 0 \ 1 0 0 2 ] in Jordan Canonical Form. We show how to find a basis that gives P. The Jordan form has 2 Jordan blocks of size 2. |

Lecture 21Play Video |
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2Matrix Theory: Find a matrix P that puts the following real 4x4 matrix A = [2 0 0 0 \ 0 2 0 0 \ 0 0 2 1 \ 1 0 0 2] into Jordan Canonical Form. Here the JCF has blocks of size 3 and 1. We focus on finding a vector that generates the 3x3 block. |

Lecture 22Play Video |
Commutant of Complex MatrixMatrix Theory: Let A be an nxn matrix with complex entries. We show that the commutant of A has dimension greater than or equal to n. The key step is to show the result for the Jordan canonical form of A. |

Lecture 23Play Video |
Example of Rational Canonical Form 1: Single BlockMatrix Theory: Let A be the real matrix [0 -1 1 0 \ 1 0 0 1 \ 0 0 0 -1 \ 0 0 1 0]. Find a matrix P that puts A into rational canonical form over the real numbers. We compare RCF with Jordan canonical form and review companion matrices. (Minor corrections added.) |

Lecture 24Play Video |
Example of Rational Canonical Form 2: Several BlocksMatrix Theory: Let A be a 12x12 real matrix with characteristic polynomial (x^2+1)^6, minimal polynomial (x^2 + 1)^3, and dim(Null(A^2 + I)) = 6. Find all possible rational canonical forms for A. |

Lecture 25Play Video |
Example of Rational Canonical Form 3Matrix Theory: We note two formulations of Rational Canonical Form. A recipe is given for combining and decomposing companion matrices using cyclic vectors. |

Lecture 26Play Video |
Exponential of 2x2 Matrix 1: Complex CaseMatrix Theory: We give a method for computing the exponential of a 2x2 matrix A with complex coefficients. We note that any such matrix has a Jordan form that is diagonal or a 2x2 Jordan block. |

Lecture 27Play Video |
Exponential of 2x2 Matrix 2: Traceless CaseMatrix Theory: We compute the exponential of a real 2x2 matrix A when the trace of A is zero. We use the Cayley-Hamilton Theorem to obtain explicit formulas based on the determinant of A. |

Lecture 28Play Video |
Example of Group ActionMatrix Theory: Consider the set G of matrices of the form [x y \ 1 0] where x is nonzero real and y is real. Let G act on the real line R by [x y \ 1 0].t = xt + y. Show that G is a group, that the action is a group action, and that the action is faithful. |

Lecture 29Play Video |
Example of QuaternionsMatrix Theory: For the quaternion alpha = 1 - i + j - k, find the norm N(alpha) and alpha^{-1}. Then write alpha as a product of a length and a direction. |

Lecture 30Play Video |
The Fibonacci Numbers Using Linear Algebra (HD Version)Linear Algebra: We derive the Binet Formula for the Fibonacci numbers using linear algebra. The technique involves using diagonalization to compute the power of a matrix. |

Lecture 31Play Video |
The Fibonacci Numbers Using Power SeriesWe revisit the Binet formula for the Fibonacci numbers as an application of generating functions. This derivation requires no linear algebra, only power series methods. |

Lecture 32Play Video |
Example of Group Automorphism 1 (Requires Linear Algebra)Matrix Theory: We compute the automorphism groups of G = Z/10 and G=Z/2 x Z/2. The first case is a warm up for Part 2. The second case can be recast as a linear algebra problem with matrix groups. |

Lecture 33Play Video |
Example of Group Automorphism 2: G = Z/4 x Z/4 (Requires Linear Algebra)Matrix Theory: Let G be the group Z/4 x Z/4. We show that the automorphism group Aut(G) is isomorphic to the set of all 2x2 matrices with entries in Z/4 that have determinant 1 or 3. We show that Aut(G) has order 96 and give a factorization method. |

Lecture 34Play Video |
Group Theory: The Simple Group of Order 168 - Part 1We present two realizations of the simple group of order 168. In part 1, we count the number of matrices in PSL(2,Z/7) and SL(3,Z/2). |

Lecture 35Play Video |
Group Theory: The Simple Group of Order 168 - Part 2We show that there are no nontrivial normal subgroups in SL(3,Z/2). Techniques include Jordan canonical forms and companion matrices. |