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Course

Subject:	Mathematics
Topic:	Linear Algebra
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Educator

Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Linear Algebra II: Matrix Theory

Video Lectures

Displaying all 35 video lectures.

Lecture 1 Play Video	Matrix Inverse over the Complex Numbers Matrix 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 2 Play Video	Cramer's Rule over the Complex Numbers Matrix 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 3 Play Video	Gaussian Elimination over Z/3 Matrix Theory: Find all solutions to the system of linear equations over the finite field Z/3: x+ 2y = 1, 2x + y = 2.
Lecture 4 Play Video	Matrix Inverse over Z/7 Matrix Theory: Solve the following system of equations over Z/7 by finding A^{-1}: 3x + 2y = 4, x + 5y = 1.
Lecture 5 Play Video	Cramer's Rule over Z/5 Matrix Theory: Using Cramer's Rule, we solve a system of linear equations Ax = b over the finite field Z/5.
Lecture 6 Play Video	Example of Skew-Symmetric Matrix Matrix 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 7 Play Video	Example of Simultaneous Diagonalization Matrix 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 8 Play Video	Positive Semi-Definite Matrix 1: Square Root Matrix 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 9 Play Video	Positive Semi-Definite Matrix 2: Spectral Theorem Matrix 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 10 Play Video	Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices Matrix 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 11 Play Video	Cayley-Hamilton Theorem for 2x2 Matrices Matrix 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 12 Play Video	Inverse of a Matrix Using the Cayley-Hamilton Theorem Matrix 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 13 Play Video	Cayley-Hamilton Theorem: General Case Matrix 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 14 Play Video	Cayley-Hamilton Theorem: Example 1 Matrix 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 15 Play Video	Cayley-Hamilton Theorem Example 2 Matrix 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 16 Play Video	Example of Invariant Subspace Matrix 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 17 Play Video	Overview of Jordan Canonical Form Matrix 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 18 Play Video	Example of Jordan Canonical Form: 2x2 Matrix Matrix 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 19 Play Video	Example of Jordan Canonical Form: General Properties Matrix 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 20 Play Video	Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1 Matrix 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 21 Play Video	Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2 Matrix 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 22 Play Video	Commutant of Complex Matrix Matrix 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 23 Play Video	Example of Rational Canonical Form 1: Single Block Matrix 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 24 Play Video	Example of Rational Canonical Form 2: Several Blocks Matrix 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 25 Play Video	Example of Rational Canonical Form 3 Matrix Theory: We note two formulations of Rational Canonical Form. A recipe is given for combining and decomposing companion matrices using cyclic vectors.
Lecture 26 Play Video	Exponential of 2x2 Matrix 1: Complex Case Matrix 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 27 Play Video	Exponential of 2x2 Matrix 2: Traceless Case Matrix 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 28 Play Video	Example of Group Action Matrix 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 29 Play Video	Example of Quaternions Matrix 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 30 Play 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 31 Play Video	The Fibonacci Numbers Using Power Series We 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 32 Play 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 33 Play 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 34 Play Video	Group Theory: The Simple Group of Order 168 - Part 1 We 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 35 Play Video	Group Theory: The Simple Group of Order 168 - Part 2 We show that there are no nontrivial normal subgroups in SL(3,Z/2). Techniques include Jordan canonical forms and companion matrices.