Let U be an upper triangular matrix with all its diagonal entries equal. Prove that U is complet

Let U be an upper triangular matrix with all its diagonal...

Key Concepts

Jordan Canonical Form

The Jordan Canonical Form is a representation of a matrix that groups eigenvalues into blocks, reflecting any failure of diagonalizability. When a square matrix with repeated eigenvalues is not diagonal, it typically has one or more Jordan blocks larger than 1×1, indicating that it lacks a full set of eigenvectors and is thus not diagonalizable.

Diagonalizability

Diagonalizability refers to the ability of a matrix to be written in a diagonal form via a similarity transformation, which is equivalent to the matrix having a complete set of linearly independent eigenvectors. If a matrix has defective eigenvalues (i.e., the algebraic multiplicity exceeds the geometric multiplicity), then it is not diagonalizable.

Diagonal Matrix

A diagonal matrix is a special type of upper triangular matrix where all off-diagonal entries are zero. In other words, only the entries on the main diagonal may be non-zero. Diagonal matrices are particularly easy to work with because many matrix operations, including inversion and raising to powers, become straightforward.

Upper Triangular Matrix

An upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero. This structure simplifies many computations, such as finding determinants and eigenvalues, since the eigenvalues are directly given by the entries on the main diagonal.

Eigenvalues of Triangular Matrices

For any triangular matrix, whether upper or lower, the eigenvalues are exactly the entries on its main diagonal. This property is significant because it offers a simple method to determine the spectrum of the matrix without complex calculations.

Transcript

00:01 For this problem, we have two matrices, a and b, and both are going to be upper triangular or they're both going to be lower triangular.

00:10 I'm going to set this up as if they're upper triangular.

00:13 The same proof applies for lower triangular as well.

00:17 So let me just write down some basic definitions and stuff here, so as we're referencing them, it kind of makes sense.

00:26 So a is a matrix whose elements are a, i, j.

00:30 A is the row, j is the column.

00:33 And b is going to be the same.

00:35 Okay.

00:36 And these are both going to be upper triangular.

00:44 In other words, everything below the diagonal are going to be zeros.

00:49 So the elements in a are going to be zero if the row is greater than the column.

00:58 Okay, those that will give me zeros below the diagonal.

01:01 And the same for b.

01:02 They're both the same type of matrix.

01:06 So again, the element.

01:07 Elements in b will be zero if the row is greater than the column.

01:12 I'm not making anything from the diagonal or above.

01:16 There could be zeros elsewhere.

01:18 We just know for sure that there's zeros below the diagonal.

01:21 Now, if i multiply these together, a times b, and that's going to give us a matrix, and we'll call these elements x that we get when we multiply them together.

01:32 Let's take a look.

01:33 We need to look at the diagonal matrix, the diagonal elements, of matrix a times b.

01:40 We want to prove that that diagonal that's made up of x11, x2, x33, and so on.

01:50 We want to show that those entries are the products of the diagonal entries of a and b.

01:57 So i want to show that x11 is a11111, x2222222, a33333 and b3333.

02:09 And so on.

02:11 Okay.

02:12 So let's see what we have.

02:16 X11, first row, first column in my product, in my solution...