Equal versus Transposed Matrices
Matrices are very handy to have around when you’re solving certain finite math problems. Matrices are simple structures that you can use to change the format of mathematical statements to make them more usable and understandable.
Two matrices are equal to one another if they are exactly the same size and have exactly the same elements in exactly the same places. You can’t get much more equal than that. And when a matrix is transposed, it keeps all its elements, but most switch positions.
The two matrices X and Y are equal to one another. They’re the same size, and their respective elements are equal.
Because X = Y, you can deduce, from the matrices shown, that x = –3, b = 1, c = 2, y = 0, and f = 4. This may not seem like a huge deal, but the property comes in very handy when dealing with matrix applications.
When you transpose a rectangular matrix, you usually change its shape. A square matrix keeps the same shape. But, in both cases, most of the elements change positions. The element that used to be in the second row, third column, gets moved to the third row, second column. A matrix that used to be 4 × 2 becomes 2 × 4 in the transformation.
The symbol to indicate a matrix transpose is a capital T at the upper-right corner of the matrix. The matrix M is transposed here:
The rows become columns, and the columns become rows.
With a square matrix, the dimension stays the same, and all the elements on the main diagonal, running from the top left to the bottom right, stay in their original position.
The matrix transpose comes in handy when you need to change the dimension of a matrix. You change it to perform certain matrix operations or to better organize your data.