Let A=( B 𝐜 0 0 ) be in block form, where B is an n ×n matrix, 𝐜 ∈ℝ^n, while 0 d

Let A=( B 𝐜 0 0 ) be in block form, where B...

Key Concepts

Matrix Exponential

The matrix exponential is a way to generalize the scalar exponential function to square matrices. It is defined via its power series, e^(tA) = I + tA + (tA)^2/2! + (tA)^3/3! + ?, and it plays a key role in solving systems of linear differential equations, among other applications.

Block Matrices

Block matrices are matrices that are partitioned into smaller matrices or blocks. In many cases, especially when one of the blocks has a particular structure (e.g., zero or triangular), its exponential or other functions can be computed by considering the blocks separately, which simplifies analysis and calculations.

Power Series Representation

Using the power series definition of the exponential, one can compute e^(tA) term by term. In the case of block matrices, one needs to understand how the multiplication of blocks works so that the series can be summed separately for each block, respecting the block structure of the matrix.

Integral Representation for the Off?Diagonal Block

For matrices in block triangular form, the off-diagonal block in the exponential can be expressed as an integral. In this case, by collecting the terms corresponding to the off-diagonal entries in the power series, one obtains a formula such as f(t) = ??? e^((t?s)B)c ds. This representation comes out naturally when grouping coefficients of the off-diagonal blocks in the series.

Transcript

00:01 We are given three matrices and we are asked to answer questions using these matrices.

00:08 These matrices are a, which is ab0a, b, which is a0 ,0 ,0, and c, which is 0, b, and c, which is 0, b, 0 .00.

00:17 And we see that the relationship between these three matrices is that a is equal to b plus c.

00:27 In part a, we are asked to show that bc is equal to c b.

00:37 So to calculate, to calculate.

00:38 B -c, this is going to be diagonal matrix entries a, a, times c, which is 0 -b -0 -0.

00:57 This is going to be 8 times 0 plus 0 -10 -1, which is 0, 8 times b plus 0 -3 -10 -0, which is ab, 0 -10 -0 plus 0 -0 times 0, and 0 -t times b plus 8 times 0, which is 0, and 0 plus 8 times 0, which is 0, 0, which is 0.

01:14 So we get the matrix 0 ab 0.

01:25 Likewise, the matrix c -b is 0b -0 -0 times a -0 -0 -0 -0 -3 -0.

01:36 This is equal to 0 times a, 3 times 0 plus b times a is b -a, and 0 plus 0 -10 -10 is 0, 0 plus 0 times a plus 0 times a is 0.

01:54 And we assume that entries a, b, lie in some field, or in a commutative ring, most likely we're just thinking of them as real numbers, so that would be more specifically a field.

02:18 Then we have that a -e is equal to b -a, and so that this implies matrix b is equal to c -b.

02:27 So we wanted to show for part a.

02:33 Part b, you're asked to calculate c squared and determine e to the c t.

02:44 So c squared is going to be 0b 0 0 times 0 0 0, which is equal to 0 plus 3 to 0 is 0.

02:59 0 .0 plus b2 0 is 0 .0 plus 0 times 0 is 0, and 0 plus 0 plus 0 plus 0 is 0.

03:09 And so we get the 0 matrix, as we wanted to show.

03:18 And we did the ct.

03:22 Well, again, for any 2x 2 -by -2 matrix, c, and for any real number of t, we know that b to the ct is well defined, even though we didn't prove this, or we're 10.

03:37 Hold this.

03:41 And so this is well defined, and this is the same as, since c is equal to 0b 0 ,0, and we know that c squared is going to be.

04:05 Okay, so going back to our relationship between a, b, and c, we have that a is an upper triangular matrix, b is a diagonal matrix.

04:19 And so one way of calculating is to solve c for a minus b.

04:36 Another way is just simply look at c, the matrix, and then characteristic polynomial of c is called p lambda.

04:49 So c is going to be negative lambda squared minus b10, which is simply 0.

05:02 And characteristic equation is when it's equal 0.

05:04 So we have that we have two eigenvalues, lambda 1 equals 0, and the lambda 2 equals 0...

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