Answer to Question #146298 in Matrix | Tensor Analysis for Promise Omiponle

Question #146298
Then n x n matrix A= [aij] is called a diagonal matrix if aij= 0 when i =/= j. Verify that the product of two n x n diagonal matrices is again a diagonal matrix, and give a simple rule for determining this product.
1
Expert's answer
2020-12-01T05:49:19-0500

let:

"A = \\begin{bmatrix}\n a_{11} & 0 & 0 & \\ldots & 0 \\\\\n 0 & a_{22} & 0 & \\ldots & 0 \\\\\n \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n 0 & \\ldots & 0 & 0 & a_{nn}\n\\end{bmatrix}"


"B = \\begin{bmatrix}\n b_{11} & 0 & 0 & \\ldots & 0 \\\\\n 0 & b_{22} & 0 & \\ldots & 0 \\\\\n \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n 0 & \\ldots & 0 & 0 & b_{nn}\n\\end{bmatrix}"


"C = \\begin{bmatrix}\n c_{11} & c_{12} & \\ldots & c_{1n} \\\\\n c_{21} & c_{22} & \\ldots & c_{2n} \\\\\n \\ldots & \\ldots & \\ldots & \\ldots \\\\\n c_{n1} & \\ldots & c_{n2} & c_{nn}\n\\end{bmatrix}"


Then:

"c_{11} = a_{11} \\cdot b_{11} + 0 \\cdot 0 + \\ldots + 0 \\cdot 0 = a_{11} \\cdot b_{11} \\\\\nc_{12} = a_{11} \\cdot 0 + 0 \\cdot b_{22} + \\ldots + 0 \\cdot 0 = 0 \\\\\n\\ldots \\\\\nc_{1n} = a_{11} \\cdot 0 + 0 \\cdot 0 + \\ldots + 0 \\cdot b_{nn}= 0 \\\\"

"c_{21} = 0 \\cdot b_{11} + a_{22} \\cdot 0 + \\ldots + 0 \\cdot 0 = 0 \\\\\nc_{22} = 0 \\cdot 0 + a_{22} \\cdot b_{22} + 0 \\cdot 0 + \\ldots + 0 \\cdot 0 = a_{22} \\cdot b_{22} \\\\\n\\ldots \\\\\nc_{2n} = 0 \\cdot 0 + a_{22} \\cdot 0 + \\ldots + 0 \\cdot 0 + 0 \\cdot b_{nn}= 0 \\\\"

"\\ldots \\\\\n\\ldots \\\\\n\\ldots \\\\"

"c_{n1} = 0 \\cdot b_{11} + 0 \\cdot 0 + \\ldots + a_{nn} \\cdot 0 = 0 \\\\\nc_{n2} = 0 \\cdot 0 + 0 \\cdot b_{22} + \\ldots + a_{nn} \\cdot 0 = 0 \\\\\n\\ldots \\\\\nc_{nn} = 0 \\cdot 0 + \\ldots + 0 \\cdot 0 + a_{nn} \\cdot b_{nn}= a_{nn} \\cdot b_{nn} \\\\"

It is shown that after multiplication of matrix A and B we receive matrix C which have non-zero diagonal elements, namely: "c_{ij} = 0, i \\ne j, c_{ii} \\ne 0,i=1,2,\\ldots,n" . So C is diagonal matrix.


And formula of multiplication of two diagonal matrices is:

"A \\cdot B = C \\\\\nc_{ii} = a_{ii} \\cdot b_{ii}, i = \\overline{1,n} \\\\\nc_{ij} = 0, i \\ne j"



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