Skew-symmetric matrices can be identified in a similar manner. Consider the following matrix:. In this case, on either side of the diagonal, the entries are the same but with the opposite signs. In particular, the diagonal entries are all zero, i.
Taking the transpose of this matrix would have no effect, since all the entries are the same. In fact, square-zero matrices are the only type of matrix that are both symmetric and skew symmetric at the same time. To check the equality of two matrices, we must check that the corresponding entries are equal. Clearly, the diagonal entries are the same, as the transpose does not change them. Let us highlight the off-diagonal entries that must be equal below:.
We can see that both equations are the same. In order for this matrix equation to be satisfied, the corresponding entries must be equal. We, first of all, note that on both sides of the equation, the diagonal entries are zero which is a necessary condition for a matrix to be skew symmetric. For the off-diagonal entries, we highlight where they must be equal:. This results in six equations, although in the cases where we have used the same highlight color, the equations are equivalent.
In our final example, we will consider how the transpose interacts with other matrix operations. The left-hand side of the given equation applies the transpose after the matrix subtraction, while the right-hand side applies the transpose to each matrix before the matrix subtraction.
Hence, this example is asking whether we can interchange the order of the transpose and the subtraction. Now, to check whether this equation is correct, we need to calculate each side of it and verify that the matrices are equal.
Recall that we can take the difference of two matrices of the same order by taking the difference of the corresponding entries in the two matrices. We also recall that we can find the transpose of a matrix by swapping its rows with its columns. This is not a symmetric matrix, so we will take the transpose by swapping its rows with its columns:.
We note that the result of the example above is not coincidental. We call this property distributivity. In this explainer, we have discussed the matrix transpose and some of its interesting properties. The transpose of the matrix A is A T and has an order of 3 x 2. The matrix that is resulting from a given matrix B after changing or reversing its rows to columns and columns to rows is called the transpose of a matrix B.
After interchanging the rows and columns, the resultant transpose of the matrix A T looks like:. After reversing the rows and columns, the resultant transpose of the matrix C T looks like:.
After reversing the rows and columns, the resultant transpose of the matrix A T looks like:. In linear algebra , the transpose of a matrix is actually an operator that flips a matrix over its diagonal by switching the row and column indices of matrix B and producing another matrix.
Transpose of a matrix B is often denoted by either B' or B T. Sometimes, they are also denoted as B tr or B t. Transpose of a matrix is used in some of the linear transformations as they reveal some of the important properties of the transformation. Let's learn about some of the important properties of the transpose of a matrix:. Let's consider an example here:. We can see from the above example that the sum remains the same in both cases.
Thus, transpose operation respects addition. A matrix is considered to be horizontal when the number of rows in the matrix is less than the number of columns in that matrix. And, a matrix is considered to be vertical, when the number of columns in the matrix is less than the number of rows in that matrix. Let's consider a horizontal matrix P and a vertical matrix Q as:. From the above two examples, we can see that the transpose of a horizontal matrix P results in a vertical matrix P T and the transpose of a vertical matrix Q results in a horizontal matrix Q T.
Consider the two given symmetric matrices A and B:. We can see from the above example that, after taking the transposes of the two matrices A and B, they are equal to their original matrices i. Consider the two given diagonal matrices C and D:. We can see from the above two examples that the two diagonal matrices C and D remain as diagonal matrices even after the transpose is applied.
As per one of the properties of the transpose of a matrix, when we apply transpose on a matrix B T that is already transposed, then the result matrix will be the same as the original matrix B.
It can be a means to get to a least-squares solution if you were looking to model, say, how much of each fruit you'd expect to sell on a particular day. This was one of the answers in the linked question. This is consistent with the abovementioned. It can be a part of a bigger question context needed, for example, linear regression as mentioned by lhf in the linked post.
John is right, in your example, it doesn't make sense. But I'll attempt to explain it in a simpler example than what Brethlosze said Let's say you are measuring something like acceleration.
But your measurement of acceleration always has an error. This error propagates as you continue your calculations. If you want to calculate velocity from that acceleration, then that acceleration error is going to propagate into the velocity calculations.
If you continue to calculate distance, than that acceleration error is going to also continue to propagate into the distance calculation. Now this is a very simple example, but it shows the relationship between this measurement and its error and all the calculates values state variables. Now imagine a more complicated scenario where you need more than 1 measurement to get the desired state variable that you are looking for.
You will have an equation with 2 variables to show the relationship between the measurements and the state variable. What if you are looking for 4 state variable that each are composed of 2 other state variables that those in itself take 2 or 3 or 4 or however many measurements to calculate. Now you can use a matrix to show the relationships between all these measurements and state variables. So now, if we transpose the matrix and multiply it by the original matrix, look at how those equations in the matrix are being multiplied with all the other variables and itself.
Try the math of a simple 2x2 times the transpose of the 2x2. This is the covariance. Wikipedia: In probability theory and statistics, covariance is a measure of the joint variability of two random variables. Too me, its like the covariance matrix mixes all the variables and measurements in every way possible to show how all of them vary against and with each other.
This is very important in inertial measurement units IMUs. There are many error states sometimes more than a dozen in an IMU in order to calculate the state variables position, velocity, body orientation, etc. These error states and state variables are defined in a matrix. When you find the covariance of the matrix, you can define the joint variability between all these error states and state variables.
This is important in order to know how these errors and measurements can affect your accuracy of position, velocity, and body orientation. By knowing your covariance, you pretty know how accurate your IMU is at defining your position, velocity, orientation, etc. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
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