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The x- and y-coordinates of the point, is its position vector. The same idea can be applied to a point in 3 dimensional space, where the position vector has 3 elements. There are many other examples of physically measurable quantities that are also mathematically represented as vectors, like velocity, acceleration, electric and magnetic fields, etc. So how do matrices fit into the picture? We already saw that elementary row operations on a matrix (or vector) can be performed by multiplying it by an appropriate special matrix.

Com 64 Click on the ad to read more Matrix Methods And Differential Equations Eigenvalues and Diagonalization First reduce this to row echelon form 1 −1 3   2 2 −2     3 1 1  by swapping row 1 and row 3 1 −1 3  0 4 −8   0 4 −8 row 2 = row 2 –2*row 1; row 3 = row 3 –3* row 1 1 0 1  0 1 −2    0 0 0  row 3 = row 3 – row 2; row 2 = row 2/4; row 1 = row 1 + row 2 In the standard way, we ignore the last row; the 2nd row allows us to choose the arbitrary value 1 for x3 and this leads to x2 = 2 and then from the first row x3 = -1.

To make it even more so, we would like to factor out the [ – but need to be careful, because the temptation is to write (A-l) for the remaining factor. That would be wrong, because A is a matrix while l is a number and we can only subtract objects that conform. e. l [ = l I [ so that we get 0 (A−λ I) x = Now the equation really does look like a homogeneous set, which we know how to solve – but there is one difference. Not only is the vector [ unknown, as before, but in addition there is an unknown l in the coefficient matrix!

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