15. Find all four square-roots of the matrix M =

15. Find all four square-roots of the matrix $M = \begin{bmatrix} -5 & 6\\ -9 & 10 \end{bmatrix}$. 16. Determine the spectral decomposition of the matrix $M = \begin{bmatrix} 1 & -4 & 0\\ -4 & 1 & 0\\ 0 & 0 & 5 \end{bmatrix}$. In other words, find the eigendecomposition of R³ for M and the corresponding orthogonal projection(s).

Transcript

00:01 Hello student here in this question from the given matrix that is p is equals to u1, u2, u3 is given as 1 over under root 2, 0 1 over under root 2 minus 1 over under root 18 4 over under root 18 1 over 19 minus 2 over 2 over 3 minus 1 over 3 and 2 over 3.

00:26 Now to find the spectral decomposition we have to multiply u1 u1 transpose that is equal to 1 over under root 2 0 1 over under 2 into matrix 1 over under 2 0 0 0 1 over 2 0 0 1 over 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 over 2 now multiply matrix u 2 u 2 transpose that is equals to minus 1 over under root 18 4 over under root 18 1 over under root 18 into into u2 transpose that is minus 1 over under root 18 4 over under root 18 1 over under root 18 and after multiplying this will get the matrix 1 over 18 minus 2 over 9 1 over 18 minus 2 over 9 8 over 9 8 over 9 minus 1 over 18 2 over 18 2 over 18 now multiply matrix u 3 and u 3 transpose that is equals to minus 2 over 3 minus 1 over 3 2 over 3 into u 3 transpose that is minus 2 over 3 minus 1 over 3 and after multiplying this will get the matrix 4 over 9 2 over 9 now matrix a can be written is minus u into u 1 u1 u1 transpose plus minus 1 1 1 1 1 2 2 transpose plus minus 1 1 1 into u 2 transpose plus minus 1 into u 2 transpose plus 17 into u3 into u3 transpose, that is equals to, a is equals to minus 1 over 2, 0 ,000, 0 ,000, minus 1 over 2, minus 1 over 2, plus minus 1 over 18, 2, 2 over 1 over 18, 2 minus 1 over 9, minus 8 over 9, minus 2 over 9, minus 2 over 9, minus 2 over 9, minus 2 over 9, 1 over 19, 1 over 8 over 9, minus 2 over 9, 1 over 9, 1 over 8 over 9, minus 1 over 18 plus 17 into 4 over 9 2 over 9 minus 4 over 9 2 over 9 2 over 9 1 over 9 minus 2 over 9 minus 2 over 9 minus 2 over 9 and after solving this we'll get the matrix a is equals to matrix 7 4 minus 8 4 minus 4 minus 4 minus 8 minus 8 minus 4, 7, that is equals to given matrix a, which is verified...

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