Linear Algebra II: Matrix Theory

Video Lectures

Displaying all 35 video lectures.
Lecture 1
Matrix Inverse over the Complex Numbers
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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
Cramer's Rule over the Complex Numbers
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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
Gaussian Elimination over Z/3
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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
Matrix Inverse over Z/7
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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
Cramer's Rule over Z/5
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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
Example of Skew-Symmetric Matrix
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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
Example of Simultaneous Diagonalization
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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
Positive Semi-Definite Matrix 1: Square Root
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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
Positive Semi-Definite Matrix 2: Spectral Theorem
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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
Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices
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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
Cayley-Hamilton Theorem for 2x2 Matrices
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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
Inverse of a Matrix Using the Cayley-Hamilton Theorem
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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
Cayley-Hamilton Theorem: General Case
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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
Cayley-Hamilton Theorem: Example 1
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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
Cayley-Hamilton Theorem Example 2
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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
Example of Invariant Subspace
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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
Overview of Jordan Canonical Form
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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
Example of Jordan Canonical Form: 2x2 Matrix
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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
Example of Jordan Canonical Form: General Properties
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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
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1
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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
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2
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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
Commutant of Complex Matrix
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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
Example of Rational Canonical Form 1: Single Block
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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
Example of Rational Canonical Form 2: Several Blocks
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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
Example of Rational Canonical Form 3
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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
Exponential of 2x2 Matrix 1: Complex Case
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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
Exponential of 2x2 Matrix 2: Traceless Case
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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
Example of Group Action
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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
Example of Quaternions
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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
The Fibonacci Numbers Using Linear Algebra (HD Version)
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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
The Fibonacci Numbers Using Power Series
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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
Example of Group Automorphism 1 (Requires Linear Algebra)
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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
Example of Group Automorphism 2: G = Z/4 x Z/4 (Requires Linear Algebra)
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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
Group Theory: The Simple Group of Order 168 - Part 1
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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
Group Theory: The Simple Group of Order 168 - Part 2
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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.